Question

set the area of a rectangle is given in the form of a trinomial expression. find the equivalent expression that shows the lengths of the two sides of the rectangle. 7. $x^{2}+9x + 8$ 8. $x^{2}-6x + 8$ 9. $x^{2}-2x - 8$ 10. $x^{2}+7x - 8$

Explanation:

Step1: Factor \(x^{2}+9x + 8\)

For a quadratic \(ax^{2}+bx + c\) (\(a = 1\), \(b=9\), \(c = 8\)), find two numbers \(m\) and \(n\) such that \(m + n=b\) and \(m\times n=c\). Here \(m = 1\), \(n=8\) since \(1+8 = 9\) and \(1\times8=8\). Then \(x^{2}+9x + 8=(x + 1)(x + 8)\)

Step2: Factor \(x^{2}-6x + 8\)

For \(a = 1\), \(b=-6\), \(c = 8\), find \(m\) and \(n\) with \(m + n=-6\) and \(m\times n = 8\). \(m=-2\), \(n=-4\) since \((-2)+(-4)=-6\) and \((-2)\times(-4)=8\). So \(x^{2}-6x + 8=(x - 2)(x - 4)\)

Step3: Factor \(x^{2}-2x-8\)

For \(a = 1\), \(b=-2\), \(c=-8\), find \(m\) and \(n\) with \(m + n=-2\) and \(m\times n=-8\). \(m = 2\), \(n=-4\) since \(2+(-4)=-2\) and \(2\times(-4)=-8\). Thus \(x^{2}-2x - 8=(x + 2)(x - 4)\)

Step4: Factor \(x^{2}+7x-8\)

For \(a = 1\), \(b = 7\), \(c=-8\), find \(m\) and \(n\) with \(m + n=7\) and \(m\times n=-8\). \(m = 8\), \(n=-1\) since \(8+(-1)=7\) and \(8\times(-1)=-8\). Hence \(x^{2}+7x-8=(x + 8)(x - 1)\)

Answer:

1. \((x + 1)(x + 8)\)
2. \((x - 2)(x - 4)\)
3. \((x + 2)(x - 4)\)
4. \((x + 8)(x - 1)\)