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Algebra 2 practice

Completing the Square — Practice

Rewrite quadratics as a perfect square plus a constant — the technique behind vertex form and the quadratic formula itself — with every step explained.

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What you'll practice

Completing the square rewrites x2+bxx^2 + bx by adding and subtracting (b2)2\left(\frac{b}{2}\right)^2, turning it into (x+b2)2\left(x + \frac{b}{2}\right)^2 minus a correction. It solves equations that don't factor and converts standard form to vertex form. These questions cover finding the completing constant, solving equations, and rewriting into vertex form.

Worked example

Solve x2+6x+2=0x^2 + 6x + 2 = 0.

  1. Move the constant: x2+6x=−2x^2 + 6x = -2.
  2. Half of 66 is 33; add 32=93^2 = 9 to both sides: x2+6x+9=7x^2 + 6x + 9 = 7.
  3. The left side is now a perfect square: (x+3)2=7(x + 3)^2 = 7.
  4. Square-root both sides: x+3=±7x + 3 = \pm\sqrt{7}, so x=−3±7x = -3 \pm \sqrt{7}.

Common mistakes

  • Adding the constant to only one side of the equation — whatever you add on the left must be added on the right.
  • Using b2b^2 instead of (b2)2\left(\frac{b}{2}\right)^2: for x2+10xx^2 + 10x, the completing constant is 2525, not 100100.
  • Forgetting the ±\pm when square-rooting: (x−2)2=5(x-2)^2 = 5 gives x=2±5x = 2 \pm \sqrt{5} — two solutions, not one.

Try a few

1. What number completes the square for x2+10x+  ?x^2 + 10x + \;?

  • 2525
  • 100100
  • 55
  • 5050

2. Solve x2−4x−1=0x^2 - 4x - 1 = 0 by completing the square.

  • x=2±5x = 2 \pm \sqrt{5}
  • x=−2±5x = -2 \pm \sqrt{5}
  • x=2±3x = 2 \pm \sqrt{3}
  • x=4±5x = 4 \pm \sqrt{5}

3. Rewrite x2+8x+3x^2 + 8x + 3 in vertex form.

  • (x+4)2−13(x+4)^2 - 13
  • (x+4)2+3(x+4)^2 + 3
  • (x+8)2−61(x+8)^2 - 61
  • (x+4)2−16(x+4)^2 - 16

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FAQ

How do you complete the square?
Take half the x-coefficient, square it, and add it (and subtract it, or add it to both sides of an equation). The x-terms then collapse into one squared binomial.
Why learn it if the quadratic formula exists?
It's how you convert to vertex form to read off a parabola's vertex — and it's the derivation the quadratic formula comes from, so it explains the why.
Is Cramzoo free?
Cramzoo is free to start — every unit opens with free skills and no account is needed. This particular skill is part of Premium ($6/month or $30/year), which unlocks every skill plus the parent progress dashboard.
How is this different from IXL?
No punishing score. A wrong answer gives you a step-by-step explanation, not a number that drops — and every question is reviewed by a human before you see it.

Related Algebra 2 skills

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Every question reviewed. Mistakes turn into explanations, never a score that punishes you for trying.

Start practicing — free

No login, no card. Open Algebra 2 and pick the tile.