Chapter 8, Problem 4PFA

To determine

To find the factor of the given expression.

Answer to Problem 4PFA

  $(2x-9)(4x+5)$

Explanation of Solution

Given:

Expression: $8x^2-26x-45$

Formula used:

Roots of the polynomial of the form $a{x}^2+bx+c=a(x-x_1)(x-x_2)$ are

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

Consider the expression $8x^2-26x-45,$

Substitute in the formula for $a=8,b=-26,c=-45$

  $\begin{array}{l}\Rightarrow x=\frac{-(-26)\pm \sqrt{{(-26)}^2-4(8)(-45)}}{2(8)}\\ \Rightarrow x=\frac{26\pm \sqrt{676+1440}}{16}\\ \Rightarrow x=\frac{26\pm \sqrt{2116}}{16}\\ \Rightarrow x=\frac{26\pm 46}{16}\end{array}$

Hence, the roots are,

  $\begin{array}{l}\Rightarrow x_1=\frac{26+46}{16}=\frac{72}{16}=\frac{9}{2}\\ \Rightarrow x_2=\frac{26-46}{16}=\frac{-20}{16}=\frac{-5}{4}\end{array}$

Therefore, the expression can be factored as

  $\begin{array}{l}\Rightarrow 8x^2-26x-45\\ \Rightarrow (x-\frac{9}{2})(x-(\frac{-5}{4}))\\ \Rightarrow (2x-9)(4x+5)\end{array}$

Conclusion:

Therefore, $(2x-9)(4x+5)$ is a factor of the given expression.