Solving Quadratic Equations by Square Roots

We recall that a linear equation having one variable is in the form of $ax+b=0$ , where a and b are constants and a ≠ 0.

A quadratic equation, on the other hand, has an x-squared $x^2$ term.

The general form for a quadratic equation is ${ax}^2+bx+c=0$ , where $a\ne 0$ .

Methods of solving quadratic equations

There are five possible ways to solve a quadratic equation in order to find the value or values for x that work to make it a true mathematical statement. These are:

• Guess and check
• Factoring
• Completing the square
• The quadratic formula
• Solving by square roots (when $b=0$ )

For now, we're going to look at the last method on the list, solving by square roots. Therefore, we're going to look at quadratic equations where $b=0$ .

Solving quadratic equations by square roots where b = 0

Again, the regular form of a quadratic equation is

${ax}^2+bx+c=0$

But if $b=0$ , we can solve the equation by putting it in the form of

$x^2=d$

for some new constant d, and taking the square root of both sides. We will count both positive and negative square roots because we want all the numbers that solve the equation properly. This is an easy method of solving quadratic equations that only works when $b=0$ .

Example 1

Solve the following quadratic equation for x.

$8x^2+4=76$

We need to put the equation into the form $x^2=d$ . The first step is to subtract 4 from each side.

$8x^2+4-4=76-4$

$8x^2=72$

Next, isolate $x^2$ by dividing both sides by 8.

$\frac{8x^2}{8}=\frac{72}{8}$

Simplify by performing the division.

$x^2=9$

Finally, take the square root of each side.

$x=3$ or $-3$ (because both $3^2$ and ${\left(-3\right)}^2$ equal 9)

These are the two solutions to the equation.

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