Square Root Method — Definition, Formula & Examples

The square root method is a technique for solving quadratic equations by isolating the squared expression on one side, then taking the square root of both sides to find the variable. It works directly when the equation can be written in the form

(x - h)^2 = k

Given an equation of the form $(x - h)^2 = k$ where $k \geq 0$ , the square root method yields the solutions $x = h \pm \sqrt{k}$ . When $k < 0$ , the equation has no real solutions. When $k = 0$ , there is exactly one solution (a double root) at $x = h$ .

Key Formula

\text{If } (x - h)^2 = k, \text{ then } x = h \pm \sqrt{k}

Where:

$x$ = the unknown variable you are solving for
$h$ = the value subtracted from x inside the squared expression
$k$ = the constant on the other side of the equation (must be ≥ 0 for real solutions)

How It Works

Start by using algebra to isolate the squared term on one side of the equation. Then take the square root of both sides, remembering to include both the positive and negative roots (the

\pm

symbol). Finally, solve for

x

by undoing any remaining operations. This method is most efficient when the equation is already in the form

x^2 = k

(x - h)^2 = k

, avoiding the need for factoring or the full quadratic formula.

Worked Example

Problem: Solve the equation

(x - 3)^2 = 25

Take the square root of both sides: Apply the square root to both sides, using ± on the right.

x - 3 = \pm\sqrt{25} = \pm 5

Solve for x: Add 3 to both sides to isolate x, giving two solutions.

x = 3 + 5 = 8 \quad \text{or} \quad x = 3 - 5 = -2

Answer:

x = 8

x = -2

Why It Matters

The square root method is the fastest way to solve quadratics that are already in squared form, which appears frequently when completing the square or working with geometric area problems. It also forms the logical foundation for deriving the quadratic formula itself.

Common Mistakes

Mistake: Forgetting the ± and writing only the positive square root.

Correction: A squared expression can equal a positive number in two ways. Always write

x - h = \pm\sqrt{k}

to capture both solutions.