Chapter 4.2, Problem 38E

To determine

To Find: The formula for the given quadratic equation by completing the squares.

Answer to Problem 38E

The roots of the quadratic equation is $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ .

Explanation of Solution

Given:

The given equation is $a{x}^2+bx+c=0$ .

Calculation:

The given equation is,

  $a{x}^2+bx+c=0$

Then,

  $\begin{array}{l}a{x}^2+bx=-c\\ x^2+\frac{b}{a}x=-\frac{c}{a}\\ x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=-\frac{c}{a}+\frac{b^2}{4a^2}\\ {\left(x+\frac{b}{2a}\right)}^2=\frac{b^2-4ac}{4a^2}\end{array}$

Solve further as,

  $\begin{array}{l}x+\frac{b}{2a}=\pm \sqrt{\frac{b^2-4ac}{4a^2}}\\ x=-\frac{b}{2a}\pm \sqrt{\frac{b^2-4ac}{4a^2}}\\ x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\end{array}$