Solving with Square Roots

≈ 25 min

When a quadratic has no $x$ term — shape $x^2 = k$ — take square roots directly. Remember: every positive number has two square roots.

x^2 = 49 \;\Rightarrow\; x = 7 \text{ or } x = -7 \qquad \text{(written } x = \pm 7\text{)}

Both work: $7^2 = 49$ and $(-7)^2 = 49$ .

The same idea unlocks bracketed squares.

(x - 3)^2 = 16 \;\Rightarrow\; x - 3 = \pm 4

x = 3 + 4 = 7 \quad \text{or} \quad x = 3 - 4 = -1

If $k$ is negative — $x^2 = -9$ — there is no real solution: no real number squares to a negative.

Solve: $x^2 = 49$ . Answer in the form x=… or x=…

Solve: $(x - 3)^2 = 16$ . Answer in the form x=… or x=…

Why does $x^2 = 25$ have two solutions?

Because 25 is odd

Because both $5^2$ and $(-5)^2$ equal 25

Because every equation has two solutions

It has only one solution: $x = 5$

Give the positive root of $x^2 = 81$ .