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Math Practice

Completing the Square Practice Problems

Practice the core move of completing the square: given x² + bx, find the number (b/2)² that turns it into a perfect-square trinomial. Even middle coefficients give whole-number answers; odd ones give a fraction over 4.

  1. 1.

    What constant completes the square for \(x^{2} - 14x\)?

  2. 2.

    What constant completes the square for \(x^{2} + 12x\)?

  3. 3.

    What constant completes the square for \(x^{2} - 10x\)?

  4. 4.

    What constant completes the square for \(x^{2} - 30x\)?

  5. 5.

    What constant completes the square for \(x^{2} + 28x\)?

  6. 6.

    What constant completes the square for \(x^{2} - 6x\)?

  7. 7.

    What constant completes the square for \(x^{2} + 22x\)?

  8. 8.

    What constant completes the square for \(x^{2} + 4x\)?

  9. 9.

    What constant completes the square for \(x^{2} - 18x\)?

  10. 10.

    What constant completes the square for \(x^{2} + 18x\)?
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Mathovia Team
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Completing the Square Practice Problems

The Completing the Square lesson covers the full solving procedure and vertex form; this practice set isolates the one move everything depends on — finding the constant that completes .

How the difficulty levels work

DifficultyWhat it drillsExampleAnswer
EasyPositive even
MediumEven , possibly negative
HardOdd — fractional answer

Easy and Medium keep even, so is a whole number and is a whole number.

Hard makes odd. Then is a fraction, and the completing constant is — an odd numerator over , already in lowest terms.

Using the practice problems

Pick a difficulty and a count. For each , enter the number that completes the square, then:

  • Check Answers grades the set.
  • Show Answers reveals every constant.
  • Print Worksheet builds a printable page.

The rules these practice problems drill

RuleStatement
The constant to add
Halve, then squarenot square, then halve
Resulting square
Sign of doesn’t change the constant; does set the sign inside the binomial

See the Completing the Square lesson for the full solving procedure and worked examples.

Common mistakes to watch for

Squaring instead of . Halve first, then square.

Reducing when is odd. An odd number squared is odd, so does not simplify.

Forgetting this is only step one for an equation. In an equation you add the constant to both sides; in an expression (vertex form) you add and subtract it in the same line.

Where to go next

Use the completing constant to solve quadratics in the Quadratic Equations Practice Problems, or skip to the pre-solved version with the Quadratic Formula Practice Problems. For the full method, including leading coefficients other than 1 and vertex form, see the Completing the Square lesson.

Frequently Asked Questions

What number completes the square for x² + bx?+

, which equals . Halve the coefficient of , then square it. For the answer is ; for it is .

How do I enter a fractional answer?+

Type it as a fraction, like 169/4 or 9/4. The Hard level always produces a fraction over 4, because an odd coefficient squared is odd and doesn't reduce.

Why does this work?+

A perfect-square trinomial expands to , so its constant is always the square of half the middle coefficient. Completing the square supplies exactly that missing constant.

Does the sign of b matter for the answer?+

No. is the same for and , so and are both completed by . The sign does matter for the binomial you write afterward: versus .

How does this connect to solving a quadratic?+

After adding to both sides of an equation, the left side becomes and you finish with the square-root property. Doing this in general derives the quadratic formula.

How many problems can I generate?+

Unlimited, freshly generated each time, free and with no sign-up.

More practice problems

Quadratic Equations Practice Problems Quadratic Formula Practice Problems Factoring Quadratics Practice Problems: Free Practice for Every Trinomial Type