Chapter 1.5, Problem 37E

Solution

To solve: The equation $a{\left(x+b\right)}^2=c$ .

The solutions are $-b\pm \frac{\sqrt{ac}}{a}$ .

Given information:

The equation is $a{\left(x+b\right)}^2=c$ .

Concept:

Use the formula $\sqrt{a\cdot a}=a$

Calculation:

First, divide each side of the equation by $a$ .

  ${\left(x+b\right)}^2=\frac{c}{a}$

Take the square root on each side.

  $\begin{array}{l}\sqrt{{\left(x+b\right)}^2}=\pm \sqrt{\frac{c}{a}}\\ x+b=\pm \sqrt{\frac{c}{a}}\end{array}$

Apply the quotient property.\

For any numbers $a,b>0,\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ .

  $x+b=\pm \frac{\sqrt{c}}{\sqrt{a}}$

Now, rationalize the denominator. For this, multiply the numerator and the denominator by $\sqrt{a}$ .

  $x+b=\pm \frac{\sqrt{c}}{\sqrt{a}}\cdot \frac{\sqrt{a}}{\sqrt{a}}$

Apply the product property.

For any numbers $a,b>0,\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}$ .

  $\begin{array}{c}x+b=\pm \frac{\sqrt{c\cdot a}}{\sqrt{a\cdot a}}\\ =\pm \frac{\sqrt{ac}}{a}\end{array}$

So,

  $x=-b\pm \frac{\sqrt{ac}}{a}$ .

Therefore, the solutions are $-b\pm \frac{\sqrt{ac}}{a}$ .