Answer :
Sure, let's go through each problem step-by-step and determine if the product is a quadratic expression.
1. [tex]\(3\left(x^2+7\right)\)[/tex]
- Expand the expression: [tex]\(3 \cdot x^2 + 3 \cdot 7 = 3x^2 + 21\)[/tex]
- This is a quadratic expression because the highest power of [tex]\(x\)[/tex] is 2.
2. [tex]\(29(s-4)\)[/tex]
- Expand the expression: [tex]\(29 \cdot s - 29 \cdot 4 = 29s - 116\)[/tex]
- This is not a quadratic expression because the highest power of [tex]\(s\)[/tex] is 1.
3. [tex]\((\omega+7)(\omega+3)\)[/tex]
- Expand the expression using the distributive property (FOIL method):
[tex]\[ (\omega + 7)(\omega + 3) = \omega^2 + 3\omega + 7\omega + 21 = \omega^2 + 10\omega + 21 \][/tex]
- This is a quadratic expression because the highest power of [tex]\(\omega\)[/tex] is 2.
4. [tex]\((x+9)(x-2)\)[/tex]
- Expand the expression using the distributive property (FOIL method):
[tex]\[ (x + 9)(x - 2) = x^2 - 2x + 9x - 18 = x^2 + 7x - 18 \][/tex]
- This is a quadratic expression because the highest power of [tex]\(x\)[/tex] is 2.
5. [tex]\((2t-1)(t+5)\)[/tex]
- Expand the expression using the distributive property:
[tex]\[ (2t - 1)(t + 5) = 2t^2 + 10t - 1t - 5 = 2t^2 + 9t - 5 \][/tex]
- This is a quadratic expression because the highest power of [tex]\(t\)[/tex] is 2.
6. [tex]\((x+4)(1+4)\)[/tex]
- Evaluate the constants first: [tex]\((x+4)(1+4) = (x+4)(5)\)[/tex]
- Expand the expression: [tex]\(x \cdot 5 + 4 \cdot 5 = 5x + 20\)[/tex]
- This is not a quadratic expression because the highest power of [tex]\(x\)[/tex] is 1.
7. [tex]\((2r-5)(2r-5)\)[/tex]
- Expand the expression as a square of a binomial:
[tex]\[ (2r - 5)^2 = (2r - 5)(2r - 5) = 4r^2 - 10r - 10r + 25 = 4r^2 - 20r + 25 \][/tex]
- This is a quadratic expression because the highest power of [tex]\(r\)[/tex] is 2.
8. [tex]\((3 - 4m)\)[/tex]
- This expression is already simplified and is not a product, so the highest power of [tex]\(m\)[/tex] is 1.
- This is not a quadratic expression because it is linear.
9. [tex]\((2h+4)(2h-7)\)[/tex]
- Expand the expression using the distributive property (FOIL method):
[tex]\[ (2h + 4)(2h - 7) = 4h^2 - 14h + 8h - 28 = 4h^2 - 6h - 28 \][/tex]
- This is a quadratic expression because the highest power of [tex]\(h\)[/tex] is 2.
10. [tex]\((8-3x)(8+3x)\)[/tex]
- Expand the expression using the difference of squares formula:
[tex]\[ (a - b)(a + b) = a^2 - b^2 \][/tex]
- Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 3x\)[/tex]:
[tex]\[ (8 - 3x)(8 + 3x) = 8^2 - (3x)^2 = 64 - 9x^2 \][/tex]
- This is a quadratic expression because the highest power of [tex]\(x\)[/tex] is 2.
In summary, the quadratic expressions are:
1. [tex]\(3x^2 + 21\)[/tex]
2. [tex]\(\omega^2 + 10\omega + 21\)[/tex]
3. [tex]\(x^2 + 7x - 18\)[/tex]
4. [tex]\(2t^2 + 9t - 5\)[/tex]
5. [tex]\(4r^2 - 20r + 25\)[/tex]
6. [tex]\(4h^2 - 6h - 28\)[/tex]
7. [tex]\(64 - 9x^2\)[/tex]
1. [tex]\(3\left(x^2+7\right)\)[/tex]
- Expand the expression: [tex]\(3 \cdot x^2 + 3 \cdot 7 = 3x^2 + 21\)[/tex]
- This is a quadratic expression because the highest power of [tex]\(x\)[/tex] is 2.
2. [tex]\(29(s-4)\)[/tex]
- Expand the expression: [tex]\(29 \cdot s - 29 \cdot 4 = 29s - 116\)[/tex]
- This is not a quadratic expression because the highest power of [tex]\(s\)[/tex] is 1.
3. [tex]\((\omega+7)(\omega+3)\)[/tex]
- Expand the expression using the distributive property (FOIL method):
[tex]\[ (\omega + 7)(\omega + 3) = \omega^2 + 3\omega + 7\omega + 21 = \omega^2 + 10\omega + 21 \][/tex]
- This is a quadratic expression because the highest power of [tex]\(\omega\)[/tex] is 2.
4. [tex]\((x+9)(x-2)\)[/tex]
- Expand the expression using the distributive property (FOIL method):
[tex]\[ (x + 9)(x - 2) = x^2 - 2x + 9x - 18 = x^2 + 7x - 18 \][/tex]
- This is a quadratic expression because the highest power of [tex]\(x\)[/tex] is 2.
5. [tex]\((2t-1)(t+5)\)[/tex]
- Expand the expression using the distributive property:
[tex]\[ (2t - 1)(t + 5) = 2t^2 + 10t - 1t - 5 = 2t^2 + 9t - 5 \][/tex]
- This is a quadratic expression because the highest power of [tex]\(t\)[/tex] is 2.
6. [tex]\((x+4)(1+4)\)[/tex]
- Evaluate the constants first: [tex]\((x+4)(1+4) = (x+4)(5)\)[/tex]
- Expand the expression: [tex]\(x \cdot 5 + 4 \cdot 5 = 5x + 20\)[/tex]
- This is not a quadratic expression because the highest power of [tex]\(x\)[/tex] is 1.
7. [tex]\((2r-5)(2r-5)\)[/tex]
- Expand the expression as a square of a binomial:
[tex]\[ (2r - 5)^2 = (2r - 5)(2r - 5) = 4r^2 - 10r - 10r + 25 = 4r^2 - 20r + 25 \][/tex]
- This is a quadratic expression because the highest power of [tex]\(r\)[/tex] is 2.
8. [tex]\((3 - 4m)\)[/tex]
- This expression is already simplified and is not a product, so the highest power of [tex]\(m\)[/tex] is 1.
- This is not a quadratic expression because it is linear.
9. [tex]\((2h+4)(2h-7)\)[/tex]
- Expand the expression using the distributive property (FOIL method):
[tex]\[ (2h + 4)(2h - 7) = 4h^2 - 14h + 8h - 28 = 4h^2 - 6h - 28 \][/tex]
- This is a quadratic expression because the highest power of [tex]\(h\)[/tex] is 2.
10. [tex]\((8-3x)(8+3x)\)[/tex]
- Expand the expression using the difference of squares formula:
[tex]\[ (a - b)(a + b) = a^2 - b^2 \][/tex]
- Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 3x\)[/tex]:
[tex]\[ (8 - 3x)(8 + 3x) = 8^2 - (3x)^2 = 64 - 9x^2 \][/tex]
- This is a quadratic expression because the highest power of [tex]\(x\)[/tex] is 2.
In summary, the quadratic expressions are:
1. [tex]\(3x^2 + 21\)[/tex]
2. [tex]\(\omega^2 + 10\omega + 21\)[/tex]
3. [tex]\(x^2 + 7x - 18\)[/tex]
4. [tex]\(2t^2 + 9t - 5\)[/tex]
5. [tex]\(4r^2 - 20r + 25\)[/tex]
6. [tex]\(4h^2 - 6h - 28\)[/tex]
7. [tex]\(64 - 9x^2\)[/tex]
