Some quadratics do not need factoring: when the equation is already a square, you isolate it and take the square root of both sides. These free printable worksheets solve x² = p, ax² = c and (x − h)² = k that way — and every one has two answers.
This is the square-root method, the companion to the factoring sheets. The equation arrives as a square, so there is nothing to factor: for x² = 49 you take the root of both sides and get x = ±7; for 3x² = 48 you divide by 3 first, then root; for (x − 2)² = 9 you root both sides to x − 2 = ±3, then add 2. The one thing to keep is the ±, because a positive number has two square roots and the answer is both.
Why this sheet works
It comes a step before factoring and stays beside it: the factoring sheets take x² + bx + c = 0, which these never print, and these take the square forms, which those never print. A child who has met both knows which method a question is asking for by its shape.
How this one works
One question from this sheet, worked through a step at a time.
Solve 4x² = 324
x² = 324 ÷ 4=81
x = ±√81=±9
x = 9orx = −9
First divide by 4 to get the square alone: x² = 324 ÷ 4 = 81.
Now take the square root of both sides: x = ±√81 = ±9.
Solve by square roots · Square and shifted formsAnswer key
1. 4x² = 324 → x = −9 or x = 92. (x + 6)² = 4 → x = −8 or x = −43. x² = 100 → x = −10 or x = 104. (x − 5)² = 36 → x = −1 or x = 115. 3x² = 363 → x = −11 or x = 116. (x + 4)² = 16 → x = −8 or x = 07. (x + 8)² = 100 → x = −18 or x = 28. (x + 9)² = 49 → x = −16 or x = −29. x² = 81 → x = −9 or x = 910. x² = 25 → x = −5 or x = 5
