Completing the Square Practice Test

About This Completing the Square Practice Test

This test develops completing the square as a general quadratic tool, not merely a rewriting trick. Questions ask students to create perfect-square trinomials, convert standard form to vertex form, solve quadratic equations, and read vertices and extreme values from completed-square expressions.

The set includes quadratics with leading coefficient 1, quadratics that require factoring the leading coefficient first, fractional coefficients, missing constants, and short applications. Explanations make the compensation step explicit so the new expression remains equivalent to the original.

Completing-the-Square Skills Covered

You will halve linear coefficients, square rational values, balance added constants, rewrite quadratics in vertex form, solve by taking square roots, find vertices and extrema, and determine parameters that create perfect-square trinomials.

How to Approach the Questions

1. Factor the leading coefficient from the quadratic and linear terms when it is not 1.
2. Take half of the coefficient of x inside the bracket and square it.
3. Add and subtract the same quantity so the expression stays equivalent.
4. Check the completed form by expanding it before choosing an answer.

Common Mistakes to Avoid

Common errors include using the full x-coefficient instead of half, forgetting to square the half-coefficient, changing the sign inside the binomial, failing to multiply the compensation by the leading coefficient, and keeping only one square-root branch when solving.

Practice Format

Use this free 20-question test for Algebra 1, Algebra 2, ACT-style review, classroom practice, or independent study. Immediate explanations are shown after each answer.