Square a sum or difference

{"questions":[{"content":"Select the correct option for converting $(-x-5)^2$ into a polynomial of standard form.[[choice-24]]","widgets":{"choice-24":{"type":"choice","options":["$x^2+10x+25$","$x^2-10x+25$","$-x^2-10x-25$","$x^2+5x+25$"],"explanations":["$(-x-5)=-(x+5)$, means $(-x-5)^2=(x+5)^2=x^2+10x+25$.","The middle term for $(x+5)^2$ is positive: $2\\cdot x\\cdot 5=10x$.","The square cannot give a negative leading coefficient: $(-x)^2=x^2$.","The middle term must be $10x$ because there is a factor $2$."],"answer":[0]}},"step":1,"explanation":"Note that $(-x-5)=-(x+5)$. When squaring, the minus sign disappears: $(-x-5)^2=(x+5)^2$. Using the formula for the square of the sum we obtain $x^2+10x+25$.","hints":["Factor out a minus sign: $(-x-5)=-(x+5)$, then apply $(a+b)^2=a^2+2ab+b^2$.","$(-x-5)^2=(x+5)^2=x^2+2\\cdot x\\cdot 5+25=x^2+10x+25$."]},{"content":"Select the correct option for converting $(-a-7)^2$ into a polynomial of standard form.[[choice-25]]","widgets":{"choice-25":{"type":"choice","options":["$a^2+14a+49$","$a^2-14a+49$","$-a^2-14a-49$","$a^2+7a+49$"],"explanations":["$(-a-7)=-(a+7)$, so the square is $(a+7)^2=a^2+14a+49$.","For the sum, the middle term is positive: $2\\cdot a\\cdot 7=14a$.","The square of $(-a)$ is equal to $a^2$, not $-a^2$.","The middle term should be $14a$, not $7a$."],"answer":[0]}},"step":1,"explanation":"The expression $(-a-7)^2$ is the square of the sum $(a+7)$, because $(-a-7)=-(a+7)$. Expand: $a^2+14a+49$.","hints":["Convert to the square of the sum: $(-a-7)^2=(a+7)^2$.","$(a+7)^2=a^2+2\\cdot a\\cdot 7+7^2=a^2+14a+49$."]},{"content":"Select the correct option for converting $(-3x-2)^2$ into a polynomial of standard form.[[choice-26]]","widgets":{"choice-26":{"type":"choice","options":["$9x^2+12x+4$","$9x^2-12x+4$","$3x^2+12x+4$","$9x^2+6x+4$"],"explanations":["$(-3x-2)=-(3x+2)$, which means the square is equal to $(3x+2)^2=9x^2+12x+4$.","The average term for the sum is positive: $2\\cdot 3x\\cdot 2=12x$.","$(3x)^2=9x^2$, not $3x^2$.","The middle term should be $12x$ because of the multiplier $2$."],"answer":[0]}},"step":1,"explanation":"Since $(-3x-2)=-(3x+2)$, then $(-3x-2)^2=(3x+2)^2$. We open it according to the formula: $9x^2+12x+4$.","hints":["Rewrite as a square of the sum: $(-3x-2)^2=(3x+2)^2$.","$(3x+2)^2=(3x)^2+2\\cdot 3x\\cdot 2+2^2=9x^2+12x+4$."]},{"content":"Select the correct option for converting $(-2m-n)^2$ into a polynomial of standard form.[[choice-27]]","widgets":{"choice-27":{"type":"choice","options":["$4m^2+4mn+n^2$","$4m^2-4mn+n^2$","$2m^2+4mn+n^2$","$4m^2+2mn+n^2$"],"explanations":["$(-2m-n)=-(2m+n)$, which means the square is equal to $(2m+n)^2=4m^2+4mn+n^2$.","When summing, the middle term has a “+” sign: $2\\cdot 2m\\cdot n=4mn$.","$(2m)^2=4m^2$, not $2m^2$.","The middle term is $4mn$, not $2mn$."],"answer":[0]}},"step":1,"explanation":"The square $(-2m-n)$ is equal to the square $(2m+n)$. Using the formula for the square of the sum we obtain $4m^2+4mn+n^2$.","hints":["Take out the “minus”: $(-2m-n)=-(2m+n)$.","$(2m+n)^2=(2m)^2+2\\cdot 2m\\cdot n+n^2=4m^2+4mn+n^2$."]},{"content":"Select the correct option for converting $(-p-4q)^2$ into a polynomial of standard form.[[choice-28]]","widgets":{"choice-28":{"type":"choice","options":["$p^2+8pq+16q^2$","$p^2-8pq+16q^2$","$p^2+4pq+16q^2$","$-p^2-8pq-16q^2$"],"explanations":["$(-p-4q)=-(p+4q)$, therefore $(-p-4q)^2=(p+4q)^2=p^2+8pq+16q^2$.","The square of the sum has a positive middle term: $2\\cdot p\\cdot 4q=8pq$.","The factor $2$ is missing in the middle term: it should be $8pq$.","The square does not make the leading term negative."],"answer":[0]}},"step":1,"explanation":"Here $(-p-4q)=-(p+4q)$, so the minus sign disappears when squaring. We get $p^2+8pq+16q^2$.","hints":["Go to $(p+4q)^2$ and use $(a+b)^2$.","$(p+4q)^2=p^2+2\\cdot p\\cdot 4q+(4q)^2=p^2+8pq+16q^2$."]},{"content":"Select the correct option for converting $(-6a-b)^2$ into a polynomial of standard form.[[choice-29]]","widgets":{"choice-29":{"type":"choice","options":["$36a^2+12ab+b^2$","$36a^2-12ab+b^2$","$6a^2+12ab+b^2$","$36a^2+6ab+b^2$"],"explanations":["$(-6a-b)=-(6a+b)$, means $(6a+b)^2=36a^2+12ab+b^2$.","If the sum is positive, the middle term is positive: $2\\cdot 6a\\cdot b=12ab$.","$(6a)^2=36a^2$, not $6a^2$.","The middle term should be $12ab$, not $6ab$."],"answer":[0]}},"step":1,"explanation":"Since $(-6a-b)=-(6a+b)$, then when squaring we get $(6a+b)^2$. Expand it and get $36a^2+12ab+b^2$.","hints":["Square the sum: $(-6a-b)^2=(6a+b)^2$.","$(6a+b)^2=(6a)^2+2\\cdot 6a\\cdot b+b^2=36a^2+12ab+b^2$."]},{"content":"Select the correct option for converting $(-t-11)^2$ into a polynomial of standard form.[[choice-30]]","widgets":{"choice-30":{"type":"choice","options":["$t^2+22t+121$","$t^2-22t+121$","$-t^2-22t-121$","$t^2+11t+121$"],"explanations":["$(-t-11)=-(t+11)$, which means the square is equal to $(t+11)^2=t^2+22t+121$.","The middle term of the sum is positive: $2\\cdot t\\cdot 11=22t$.","$(-t)^2=t^2$, the leading term cannot become negative.","The middle term should be $22t$, not $11t$."],"answer":[0]}},"step":1,"explanation":"The expression $(-t-11)^2$ is equal to $(t+11)^2$. We expand using the formula for the square of the sum: $t^2+22t+121$.","hints":["Convert to $(t+11)^2$.","$(t+11)^2=t^2+2\\cdot t\\cdot 11+11^2=t^2+22t+121$."]},{"content":"Select the correct option for converting $(-0{,}5y-3)^2$ into a polynomial of standard form.[[choice-31]]","widgets":{"choice-31":{"type":"choice","options":["$0{,}25y^2+3y+9$","$0{,}25y^2-3y+9$","$0{,}5y^2+3y+9$","$0{,}25y^2+1{,}5y+9$"],"explanations":["$(-0{,}5y-3)=-(0{,}5y+3)$, which means the square is equal to $(0{,}5y+3)^2=0{,}25y^2+3y+9$.","Middle term: $2\\cdot 0{,}5y\\cdot 3=3y$, “+” sign.","$(0{,}5y)^2=0{,}25y^2$, not $0{,}5y^2$.","The middle term should be $3y$ because of the multiplier $2$."],"answer":[0]}},"step":1,"explanation":"The minus before the parentheses disappears when squaring: $(-0{,}5y-3)^2=(0{,}5y+3)^2$. Next we open: $0{,}25y^2+3y+9$.","hints":["Rewrite as $(0{,}5y+3)^2$ and expand using the formula of the square of the sum.","$(0{,}5y+3)^2=(0{,}5y)^2+2\\cdot 0{,}5y\\cdot 3+3^2=0{,}25y^2+3y+9$."]},{"content":"Select the correct option for converting $(-4x-5)^2$ into a polynomial of standard form.[[choice-32]]","widgets":{"choice-32":{"type":"choice","options":["$16x^2+40x+25$","$16x^2-40x+25$","$4x^2+40x+25$","$16x^2+20x+25$"],"explanations":["$(-4x-5)=-(4x+5)$, means $(4x+5)^2=16x^2+40x+25$.","Average term for the sum: $2\\cdot 4x\\cdot 5=40x$, sign “+”.","$(4x)^2=16x^2$, not $4x^2$.","The middle term should be $40x$, not $20x$."],"answer":[0]}},"step":1,"explanation":"Since $(-4x-5)=-(4x+5)$, then the square is equal to $(4x+5)^2$. Expanding the square of the sum, we get $16x^2+40x+25$.","hints":["Reduce to $(4x+5)^2$.","$(4x+5)^2=16x^2+2\\cdot 4x\\cdot 5+25=16x^2+40x+25$."]},{"content":"Select the correct option for converting $(-7m-2n)^2$ into a polynomial of standard form.[[choice-33]]","widgets":{"choice-33":{"type":"choice","options":["$49m^2+28mn+4n^2$","$49m^2-28mn+4n^2$","$7m^2+28mn+4n^2$","$49m^2+14mn+4n^2$"],"explanations":["$(-7m-2n)=-(7m+2n)$, means $(7m+2n)^2=49m^2+28mn+4n^2$.","The middle term of the sum is positive: $2\\cdot 7m\\cdot 2n=28mn$.","$(7m)^2=49m^2$, not $7m^2$.","The middle term should be $28mn$, not $14mn$."],"answer":[0]}},"step":1,"explanation":"Squaring removes the overall minus: $(-7m-2n)^2=(7m+2n)^2$. Next, use the formula for the square of the sum: $49m^2+28mn+4n^2$.","hints":["Go to the square of the sum: $(-7m-2n)^2=(7m+2n)^2$.","$(7m+2n)^2=(7m)^2+2\\cdot 7m\\cdot 2n+(2n)^2=49m^2+28mn+4n^2$."]},{"content":"Select the correct option for converting $(-c-0{,}2)^2$ into a polynomial of standard form.[[choice-34]]","widgets":{"choice-34":{"type":"choice","options":["$c^2+0{,}4c+0{,}04$","$c^2-0{,}4c+0{,}04$","$c^2+0{,}2c+0{,}04$","$-c^2-0{,}4c-0{,}04$"],"explanations":["$(-c-0{,}2)=-(c+0{,}2)$, which means the square is equal to $(c+0{,}2)^2=c^2+0{,}4c+0{,}04$.","The middle term of the square of the sum: $2\\cdot c\\cdot 0{,}2=0{,}4c$.","The $2$ factor in the middle term is missing.","A square does not make the entire expression negative."],"answer":[0]}},"step":1,"explanation":"Since $(-c-0{,}2)=-(c+0{,}2)$, then $(-c-0{,}2)^2=(c+0{,}2)^2$. Expand: $c^2+0{,}4c+0{,}04$.","hints":["Write as $(c+0{,}2)^2$ and apply the formula $(a+b)^2$.","$(c+0{,}2)^2=c^2+2\\cdot c\\cdot 0{,}2+0{,}2^2=c^2+0{,}4c+0{,}04$."]},{"content":"Select the correct option for converting $(-5p-3q)^2$ into a polynomial of standard form.[[choice-35]]","widgets":{"choice-35":{"type":"choice","options":["$25p^2+30pq+9q^2$","$25p^2-30pq+9q^2$","$5p^2+30pq+9q^2$","$25p^2+15pq+9q^2$"],"explanations":["$(-5p-3q)=-(5p+3q)$, means $(5p+3q)^2=25p^2+30pq+9q^2$.","The average term for the sum is positive: $2\\cdot 5p\\cdot 3q=30pq$.","$(5p)^2=25p^2$, not $5p^2$.","The middle term should be $30pq$, not $15pq$."],"answer":[0]}},"step":1,"explanation":"The overall minus sign disappears when the expression is squared: $(-5p-3q)^2=(5p+3q)^2$. Expand the square of the sum: $25p^2+30pq+9q^2$.","hints":["Reduce to $(5p+3q)^2$.","$(5p+3q)^2=(5p)^2+2\\cdot 5p\\cdot 3q+(3q)^2=25p^2+30pq+9q^2$."]},{"content":"Select the correct option for converting $(-u-9v)^2$ into a polynomial of standard form.[[choice-36]]","widgets":{"choice-36":{"type":"choice","options":["$u^2+18uv+81v^2$","$u^2-18uv+81v^2$","$u^2+9uv+81v^2$","$-u^2-18uv-81v^2$"],"explanations":["$(-u-9v)=-(u+9v)$, which means the square is equal to $(u+9v)^2=u^2+18uv+81v^2$.","The square of the sum has a positive middle term: $2\\cdot u\\cdot 9v=18uv$.","The $2$ factor in the middle term is missing.","The square does not give negative coefficients for all terms at once."],"answer":[0]}},"step":1,"explanation":"Since $(-u-9v)=-(u+9v)$, when squaring we get $(u+9v)^2$. Expand: $u^2+18uv+81v^2$.","hints":["Rewrite as $(u+9v)^2$.","$(u+9v)^2=u^2+2\\cdot u\\cdot 9v+(9v)^2=u^2+18uv+81v^2$."]},{"content":"Select the correct option for converting $(-1-2x)^2$ into a polynomial of standard form.[[choice-37]]","widgets":{"choice-37":{"type":"choice","options":["$4x^2+4x+1$","$4x^2-4x+1$","$x^2+4x+1$","$4x^2+2x+1$"],"explanations":["$(-1-2x)=-(1+2x)$, which means the square is equal to $(1+2x)^2=4x^2+4x+1$.","The middle term of the sum is positive: $2\\cdot 1\\cdot 2x=4x$.","$(2x)^2=4x^2$, not $x^2$.","The middle term should be $4x$, not $2x$."],"answer":[0]}},"step":1,"explanation":"The expression $(-1-2x)^2$ is equal to $(1+2x)^2$. Using the formula for the square of the sum we obtain $4x^2+4x+1$.","hints":["Reverse the order: $(-1-2x)=-(1+2x)$ and use $(a+b)^2$.","$(1+2x)^2=1^2+2\\cdot 1\\cdot 2x+(2x)^2=1+4x+4x^2=4x^2+4x+1$."]}],"mix":1}

{"questions":[{"content":"Select the correct option for converting $(6-a)^2$ into a polynomial of standard form.[[choice-1]]","widgets":{"choice-1":{"type":"choice","options":["$36-12a+a^2$","$36+12a+a^2$","$a^2-12a-36$","$36+12a-a^2$"],"explanations":["Opening according to the formula $(a-b)^2=a^2-2ab+b^2$.","The sign of the middle term must be minus.","The constant term must be positive: $6^2=36$.","When the difference is squared, the coefficient at $a^2$ is always positive."],"answer":[0]}},"step":1,"hints":["Use the squared difference formula:<br>$(a-b)^2=a^2-2ab+b^2$.","$(6-a)^2=6^2-2\\cdot 6\\cdot a + a^2=36-12a+a^2$."]},{"content":"Select the correct option for converting $(x-9)^2$ into a polynomial of standard form.[[choice-2]]","widgets":{"choice-2":{"type":"choice","options":["$x^2-18x+81$","$x^2+18x+81$","$x^2-18x-81$","$81-9x+x^2$"],"explanations":["The middle term is $-2\\cdot x\\cdot 9=-18x$, the constant term is $9^2=81$.","This is the formula for the square of a sum.","The constant term of a square is always non-negative; there should be $+81$.","The middle term should be $-18x$, not $-9x$."],"answer":[0]}},"step":1,"hints":["Apply the formula $(a-b)^2=a^2-2ab+b^2$ at $a=x$, $b=9$.","$(x-9)^2=x^2-2\\cdot x\\cdot 9+9^2=x^2-18x+81$."]},{"content":"Select the correct option for converting $(5y-2)^2$ into a polynomial of standard form.[[choice-3]]","widgets":{"choice-3":{"type":"choice","options":["$25y^2-20y+4$","$25y^2+20y+4$","$10y^2-20y+4$","$25y^2-4y+4$"],"explanations":["$ (5y)^2=25y^2$, middle term $-2\\cdot 5y\\cdot 2=-20y$, $2^2=4$.","The sign of the middle term must be minus.","The square of the first term was calculated incorrectly: it should be $25y^2$.","The middle term should be $-20y$, not $-4y$."],"answer":[0]}},"step":1,"hints":["Consider $a=5y$, $b=2$ and the formula $(a-b)^2=a^2-2ab+b^2$.","$(5y-2)^2=(5y)^2-2\\cdot 5y\\cdot 2+2^2=25y^2-20y+4$."]},{"content":"Select the correct option for converting $(3a-7)^2$ into a polynomial of standard form.[[choice-4]]","widgets":{"choice-4":{"type":"choice","options":["$9a^2-42a+49$","$9a^2+42a+49$","$6a^2-42a+49$","$9a^2-21a+49$"],"explanations":["$ (3a)^2=9a^2$, middle term $-2\\cdot 3a\\cdot 7=-42a$, $7^2=49$.","This is the square of the sum.","The square of the first term is incorrect: it should be $9a^2$.","The middle term should be $-42a$."],"answer":[0]}},"step":1,"hints":["Substitute $a_1=3a$, $b_1=7$ into the squared difference formula.","$(3a-7)^2=9a^2-42a+49$."]},{"content":"Select the correct option for converting $(2x-11)^2$ into a polynomial of standard form.[[choice-5]]","widgets":{"choice-5":{"type":"choice","options":["$4x^2-44x+121$","$4x^2+44x+121$","$2x^2-44x+121$","$4x^2-22x+121$"],"explanations":["$ (2x)^2=4x^2$, middle term $-2\\cdot 2x\\cdot 11=-44x$, $11^2=121$.","The sign of the middle term must be minus.","The square of $2x$ is equal to $4x^2$, not $2x^2$.","The middle term should be $-44x$."],"answer":[0]}},"step":1,"hints":["Use $(a-b)^2=a^2-2ab+b^2$ with $a=2x$, $b=11$.","$(2x-11)^2=4x^2-44x+121$."]},{"content":"Select the correct option for converting $(4m-3n)^2$ into a polynomial of standard form.[[choice-6]]","widgets":{"choice-6":{"type":"choice","options":["$16m^2-24mn+9n^2$","$16m^2+24mn+9n^2$","$8m^2-24mn+9n^2$","$16m^2-12mn+9n^2$"],"explanations":["The middle term is $-2\\cdot 4m\\cdot 3n=-24mn$.","This is the square of the sum.","The square of $4m$ is equal to $16m^2$.","The coefficient for $mn$ should be $24$, not $12$."],"answer":[0]}},"step":1,"hints":["Take $a=4m$, $b=3n$ in the squared difference formula.","$(4m-3n)^2=(4m)^2-2\\cdot 4m\\cdot 3n+(3n)^2=16m^2-24mn+9n^2$."]},{"content":"Select the correct option for converting $(p-2q)^2$ into a polynomial of standard form.[[choice-7]]","widgets":{"choice-7":{"type":"choice","options":["$p^2-4pq+4q^2$","$p^2+4pq+4q^2$","$p^2-2pq+4q^2$","$p^2-4pq+2q^2$"],"explanations":["Middle term: $-2\\cdot p\\cdot 2q=-4pq$, last: $(2q)^2=4q^2$.","This is the square of the sum.","The middle term should be $-4pq$.","The last term must be $4q^2$."],"answer":[0]}},"step":1,"hints":["Apply the formula $(a-b)^2=a^2-2ab+b^2$ at $a=p$, $b=2q$.","$(p-2q)^2=p^2-4pq+4q^2$."]},{"content":"Select the correct option for converting $(7-2x)^2$ into a polynomial of standard form.[[choice-8]]","widgets":{"choice-8":{"type":"choice","options":["$4x^2-28x+49$","$49-28x+2x^2$","$4x^2+28x+49$","$49-14x+4x^2$"],"explanations":["If we count as $(7-2x)^2$, then $7^2=49$, the middle term $-2\\cdot 7\\cdot 2x=-28x$, $(2x)^2=4x^2$. In standard form: $4x^2-28x+49$.","The square of $2x$ is equal to $4x^2$, not $2x^2$.","The sign of the middle term must be minus.","The middle term should be $-28x$, not $-14x$."],"answer":[0]}},"step":1,"hints":["Consider $a=7$, $b=2x$ and the formula $(a-b)^2=a^2-2ab+b^2$.","$(7-2x)^2=49-28x+4x^2=4x^2-28x+49$."]},{"content":"Select the correct option for converting $(0{,}5x-8)^2$ into a polynomial of standard form.[[choice-9]]","widgets":{"choice-9":{"type":"choice","options":["$0{,}25x^2-8x+64$","$0{,}25x^2+8x+64$","$0{,}5x^2-8x+64$","$0{,}25x^2-4x+64$"],"explanations":["$(0{,}5x)^2=0{,}25x^2$, middle term $-2\\cdot 0{,}5x\\cdot 8=-8x$, $8^2=64$.","This is the square of the sum.","The square of $0{,}5x$ is equal to $0{,}25x^2$.","The middle term should be $-8x$."],"answer":[0]}},"step":1,"hints":["Use the formula $(a-b)^2=a^2-2ab+b^2$ for $a=0{,}5x$, $b=8$.","$(0{,}5x-8)^2=0{,}25x^2-8x+64$."]},{"content":"Select the correct option for converting $(a-\\frac{1}{3})^2$ into a polynomial of standard form.[[choice-10]]","widgets":{"choice-10":{"type":"choice","options":["$a^2-\\frac{2}{3}a+\\frac{1}{9}$","$a^2+\\frac{2}{3}a+\\frac{1}{9}$","$a^2-\\frac{1}{3}a+\\frac{1}{9}$","$a^2-\\frac{2}{3}a+\\frac{1}{3}$"],"explanations":["Middle term: $-2\\cdot a\\cdot \\frac{1}{3}=-\\frac{2}{3}a$, last: $(\\frac{1}{3})^2=\\frac{1}{9}$.","This is the square of the sum.","The coefficient of the middle term should be $\\frac{2}{3}$.","The last term must be $\\frac{1}{9}$."],"answer":[0]}},"step":1,"hints":["Substitute $a_1=a$, $b_1=\\frac{1}{3}$ into the squared difference formula.","$(a-\\frac{1}{3})^2=a^2-\\frac{2}{3}a+\\frac{1}{9}$."]},{"content":"Select the correct option for converting $(2a-\\frac{3}{2})^2$ into a polynomial of standard form.[[choice-11]]","widgets":{"choice-11":{"type":"choice","options":["$4a^2-6a+\\frac{9}{4}$","$4a^2+6a+\\frac{9}{4}$","$2a^2-6a+\\frac{9}{4}$","$4a^2-3a+\\frac{9}{4}$"],"explanations":["Middle term: $-2\\cdot 2a\\cdot \\frac{3}{2}=-6a$, last: $(\\frac{3}{2})^2=\\frac{9}{4}$.","This is the square of the sum.","The square of $2a$ is equal to $4a^2$.","The middle term should be $-6a$."],"answer":[0]}},"step":1,"hints":["Take $a_1=2a$, $b_1=\\frac{3}{2}$ and use $(a-b)^2=a^2-2ab+b^2$.","$(2a-\\frac{3}{2})^2=4a^2-6a+\\frac{9}{4}$."]},{"content":"Select the correct option for converting $(x-\\frac{y}{2})^2$ into a polynomial of standard form.[[choice-12]]","widgets":{"choice-12":{"type":"choice","options":["$x^2-xy+\\frac{y^2}{4}$","$x^2+xy+\\frac{y^2}{4}$","$x^2-\\frac{xy}{2}+\\frac{y^2}{4}$","$x^2-xy+\\frac{y^2}{2}$"],"explanations":["Middle term: $-2\\cdot x\\cdot \\frac{y}{2}=-xy$, last: $(\\frac{y}{2})^2=\\frac{y^2}{4}$.","This is the square of the sum.","The middle term should be $-xy$, not $-\\frac{xy}{2}$.","The square of $\\frac{y}{2}$ is equal to $\\frac{y^2}{4}$."],"answer":[0]}},"step":1,"hints":["Apply the formula $(a-b)^2=a^2-2ab+b^2$ at $a=x$, $b=\\frac{y}{2}$.","$(x-\\frac{y}{2})^2=x^2-xy+\\frac{y^2}{4}$."]},{"content":"Select the correct option for converting $(3x-0{,}2)^2$ into a polynomial of standard form.[[choice-13]]","widgets":{"choice-13":{"type":"choice","options":["$9x^2-1{,}2x+0{,}04$","$9x^2+1{,}2x+0{,}04$","$3x^2-1{,}2x+0{,}04$","$9x^2-0{,}6x+0{,}04$"],"explanations":["Middle term: $-2\\cdot 3x\\cdot 0{,}2=-1{,}2x$, last: $0{,}2^2=0{,}04$.","This is the square of the sum.","The square of $3x$ is equal to $9x^2$.","The middle term should be $-1{,}2x$."],"answer":[0]}},"step":1,"hints":["Take $a=3x$, $b=0{,}2$ and expand the square of the difference.","$(3x-0{,}2)^2=9x^2-1{,}2x+0{,}04$."]},{"content":"Select the correct option for converting $(a-4b)^2$ into a polynomial of standard form.[[choice-14]]","widgets":{"choice-14":{"type":"choice","options":["$a^2-8ab+16b^2$","$a^2+8ab+16b^2$","$a^2-4ab+16b^2$","$a^2-8ab+8b^2$"],"explanations":["Middle term: $-2\\cdot a\\cdot 4b=-8ab$, last: $(4b)^2=16b^2$.","This is the square of the sum.","The middle term should be $-8ab$.","The square of $4b$ is equal to $16b^2$."],"answer":[0]}},"step":1,"hints":["Use the formula $(a-b)^2=a^2-2ab+b^2$ for $b=4b$.","$(a-4b)^2=a^2-8ab+16b^2$."]},{"content":"Select the correct option for converting $(10t-s)^2$ into a polynomial of standard form.[[choice-15]]","widgets":{"choice-15":{"type":"choice","options":["$100t^2-20ts+s^2$","$100t^2+20ts+s^2$","$10t^2-20ts+s^2$","$100t^2-10ts+s^2$"],"explanations":["Middle term: $-2\\cdot 10t\\cdot s=-20ts$, first: $(10t)^2=100t^2$.","This is the square of the sum.","The square of $10t$ is equal to $100t^2$.","The middle term should be $-20ts$."],"answer":[0]}},"step":1,"hints":["Consider $a=10t$, $b=s$ and apply the squared difference formula.","$(10t-s)^2=(10t)^2-2\\cdot 10t\\cdot s+s^2=100t^2-20ts+s^2$."]}]}

{"questions":[{"content":"Select the correct option for converting $(8+b)^2$ into a polynomial of standard form.[[choice-12]]","widgets":{"choice-12":{"type":"choice","options":["$64+16b+b^2$","$16+64b+b^2$","$64+8b^2+2b$","$64-16b+b^2$"],"explanations":["Expanding using the formula $(a+b)^2=a^2+2ab+b^2$ gives $64+16b+b^2$.","The square of 8 is equal to 64, not 16; the coefficient for $b$ is also not the same.","The coefficient for $b$ should be $16b$, not $2b$; The terms are also mixed up.","Here we would get the formula for the squared difference, but the sign should be “+”."],"answer":[0]}},"step":1,"hints":["Use the formula for the square of the sum:<br />$(a+b)^2=a^2+2ab+b^2$.","$(8+b)^2=8^2+2\\cdot 8\\cdot b +b^2=64+16b+b^2$."]},{"content":"Select the correct option for converting $(x+5)^2$ into a polynomial of standard form.[[choice-13]]","widgets":{"choice-13":{"type":"choice","options":["$x^2+10x+25$","$x^2+5x+25$","$x^2-10x+25$","$x^2+10x-25$"],"explanations":["According to the formula $(x+5)^2=x^2+2\\cdot x\\cdot 5+25$.","The factor 2 in the middle term is missing.","This is the expansion of $(x-5)^2$.","The last term must be $+25$, since $5^2=25$."],"answer":[0]}},"step":1,"hints":["$(a+b)^2=a^2+2ab+b^2$.","$(x+5)^2=x^2+2\\cdot x\\cdot 5+5^2=x^2+10x+25$."]},{"content":"Select the correct option for converting $(2y+3)^2$ into a polynomial of standard form.[[choice-14]]","widgets":{"choice-14":{"type":"choice","options":["$4y^2+12y+9$","$4y^2+6y+9$","$2y^2+12y+9$","$4y^2-12y+9$"],"explanations":["$(2y)^2=4y^2$, middle term: $2\\cdot (2y)\\cdot 3=12y$, and $3^2=9$.","The factor 2 was forgotten in the middle term.","The expression $2y$ was squared incorrectly.","The middle term must have a “+” sign because a sum is being squared."],"answer":[0]}},"step":1,"hints":["If $a=2y$, $b=3$, then $(a+b)^2=a^2+2ab+b^2$.","$(2y+3)^2=(2y)^2+2\\cdot(2y)\\cdot 3+3^2=4y^2+12y+9$."]},{"content":"Select the correct option for converting $(3a+1)^2$ into a polynomial of standard form.[[choice-15]]","widgets":{"choice-15":{"type":"choice","options":["$9a^2+6a+1$","$9a^2+3a+1$","$6a^2+6a+1$","$9a^2-6a+1$"],"explanations":["$(3a)^2=9a^2$, middle term $2\\cdot 3a\\cdot 1=6a$.","The middle term should be $6a$.","Incorrectly squared $3a$.","The sign of the middle term must be “+”."],"answer":[0]}},"step":1,"hints":["$(3a+1)^2=(3a)^2+2\\cdot 3a\\cdot 1+1^2$.","We get $9a^2+6a+1$."]},{"content":"Select the correct option for converting $(m+7)^2$ into a polynomial of standard form.[[choice-16]]","widgets":{"choice-16":{"type":"choice","options":["$m^2+14m+49$","$m^2+7m+49$","$m^2-14m+49$","$m^2+14m-49$"],"explanations":["The middle term is $2\\cdot m\\cdot 7=14m$, the last one is $49$.","The 2 multiplier is missing here.","This is the formula for the square of the difference $(m-7)^2$.","The last term is always positive: $7^2=49$."],"answer":[0]}},"step":1,"hints":["$(m+7)^2=m^2+2\\cdot m\\cdot 7+7^2$.","$m^2+14m+49$."]},{"content":"Select the correct option for converting $(4x+5)^2$ into a polynomial of standard form.[[choice-17]]","widgets":{"choice-17":{"type":"choice","options":["$16x^2+40x+25$","$16x^2+20x+25$","$8x^2+40x+25$","$16x^2-40x+25$"],"explanations":["$(4x)^2=16x^2$, middle term $2\\cdot 4x\\cdot 5=40x$, $5^2=25$.","The factor 2 was forgotten in the middle term.","$(4x)^2$ was calculated incorrectly.","The sign of the middle term must be “+”."],"answer":[0]}},"step":1,"hints":["Consider $a=4x$, $b=5$.","$(4x+5)^2=16x^2+40x+25$."]},{"content":"Select the correct option for converting $(p+2q)^2$ into a polynomial of standard form.[[choice-18]]","widgets":{"choice-18":{"type":"choice","options":["$p^2+4pq+4q^2$","$p^2+2pq+4q^2$","$p^2-4pq+4q^2$","$p^2+4pq+2q^2$"],"explanations":["Middle term: $2\\cdot p\\cdot 2q=4pq$, second square: $(2q)^2=4q^2$.","The middle term was found incorrectly.","The sign of the middle term corresponds to the square of the difference.","Incorrectly squared $2q$."],"answer":[0]}},"step":1,"hints":["$(p+2q)^2=p^2+2\\cdot p\\cdot 2q+(2q)^2$.","We get $p^2+4pq+4q^2$."]},{"content":"Select the correct option for converting $(6+t)^2$ into a polynomial of standard form.[[choice-19]]","widgets":{"choice-19":{"type":"choice","options":["$t^2+12t+36$","$t^2+6t+36$","$t^2-12t+36$","$36+t$"],"explanations":["$(t+6)^2=t^2+2\\cdot t\\cdot 6+36$.","There is no factor 2 in the middle term.","This is the expansion of $(t-6)^2$.","The square of the sum always gives a polynomial of the second degree."],"answer":[0]}},"step":1,"hints":["Rearranging the terms does not change the sum: $(6+t)^2=(t+6)^2$.","$(t+6)^2=t^2+12t+36$."]},{"content":"Select the correct option for converting $(0{,}5x+2)^2$ into a polynomial of standard form.[[choice-20]]","widgets":{"choice-20":{"type":"choice","options":["$0{,}25x^2+2x+4$","$0{,}25x^2+x+4$","$0{,}5x^2+2x+4$","$0{,}25x^2-2x+4$"],"explanations":["$(0{,}5x)^2=0{,}25x^2$, the middle term of $2\\cdot 0{,}5x\\cdot 2=2x$, and $2^2=4$.","The average term was calculated incorrectly.","Incorrectly squared $0{,}5x$.","The sign of the middle term must be “+”."],"answer":[0]}},"step":1,"hints":["Let $a=0{,}5x$, $b=2$. Then $(a+b)^2=a^2+2ab+b^2$.","$(0{,}5x+2)^2=0{,}25x^2+2x+4$."]},{"content":"Select the correct option for converting $\\left(\\frac{x}{3}+9\\right)^2$ into a polynomial of standard form.[[choice-21]]","widgets":{"choice-21":{"type":"choice","options":["$\\frac{x^2}{9}+6x+81$","$\\frac{x^2}{3}+6x+81$","$\\frac{x^2}{9}+3x+81$","$\\frac{x^2}{9}-6x+81$"],"explanations":["Middle term: $2\\cdot \\frac{x}{3}\\cdot 9=6x$, square 9 is equal to 81.","The square $\\left(\\frac{x}{3}\\right)^2$ was found incorrectly.","The middle term should be $6x$.","The sign of the middle term must be “+”."],"answer":[0]}},"step":1,"hints":["$\\left(\\frac{x}{3}+9\\right)^2=\\left(\\frac{x}{3}\\right)^2+2\\cdot \\frac{x}{3}\\cdot 9+9^2$.","We get $\\frac{x^2}{9}+6x+81$."]},{"content":"Select the correct option for converting $(2a+3b)^2$ into a polynomial of standard form.[[choice-22]]","widgets":{"choice-22":{"type":"choice","options":["$4a^2+12ab+9b^2$","$4a^2+6ab+9b^2$","$2a^2+12ab+3b^2$","$4a^2-12ab+9b^2$"],"explanations":["Middle term: $2\\cdot (2a)\\cdot (3b)=12ab$.","The middle term lacks the factor 2.","The terms were squared and/or multiplied incorrectly.","This would be for $(2a-3b)^2$."],"answer":[0]}},"step":1,"hints":["Let $a_1=2a$, $b_1=3b$. Then $(a_1+b_1)^2=a_1^2+2a_1b_1+b_1^2$.","$(2a+3b)^2=4a^2+12ab+9b^2$."]},{"content":"Select the correct option for converting $(k+1)^2$ into a polynomial of standard form.[[choice-23]]","widgets":{"choice-23":{"type":"choice","options":["$k^2+2k+1$","$k^2+k+1$","$k^2-2k+1$","$2k^2+2k+1$"],"explanations":["Middle term $2\\cdot k\\cdot 1=2k$.","The middle term should be $2k$.","This is the expansion of the squared difference.","The square $k$ was found incorrectly."],"answer":[0]}},"step":1,"hints":["$(k+1)^2=k^2+2k+1$.","Check that the middle term is $2k$."]},{"content":"Select the correct option for converting $(5y+0{,}2)^2$ into a polynomial of standard form.[[choice-24]]","widgets":{"choice-24":{"type":"choice","options":["$25y^2+2y+0{,}04$","$25y^2+y+0{,}04$","$5y^2+2y+0{,}04$","$25y^2-2y+0{,}04$"],"explanations":["Middle term: $2\\cdot 5y\\cdot 0{,}2=2y$, square $0{,}2$ is equal to $0{,}04$.","The middle term should be $2y$.","Incorrectly squared $5y$.","The sign of the middle term must be “+”."],"answer":[0]}},"step":1,"hints":["Use $(a+b)^2=a^2+2ab+b^2$ with $a=5y$, $b=0{,}2$.","$(5y+0{,}2)^2=25y^2+2y+0{,}04$."]},{"content":"Select the correct option for converting $(3-2x)^2$ into a polynomial of standard form.[[choice-25]]","widgets":{"choice-25":{"type":"choice","options":["$4x^2-12x+9$","$4x^2+12x+9$","$9-4x^2$","$4x^2-6x+9$"],"explanations":["This is the square of the difference: $(3-2x)^2=3^2-2\\cdot 3\\cdot 2x+(2x)^2=9-12x+4x^2$, in standard form $4x^2-12x+9$.","The sign of the middle term must be “−”, since there is a difference in parentheses.","When expanded, all three terms appear, not two.","The middle term is $-12x$, not $-6x$."],"answer":[0]}},"step":1,"hints":["Convert the expression to sum form: $(3-2x)^2=(3+(-2x))^2$ and apply $(a+b)^2=a^2+2ab+b^2$.","$3^2+2\\cdot 3\\cdot (-2x)+(-2x)^2=9-12x+4x^2=4x^2-12x+9$."]}]}