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Algebra 2 practice

Completing the Square

To complete the square on x2+bxx^2 + bx, take half of bb and square it: for x2+6xx^2 + 6x, half of 66 is 33, add 99, and it becomes (x+3)2(x+3)^2. It's the technique behind vertex form and the quadratic formula itself. Below: the method step by step, the (b2)2\left(\frac{b}{2}\right)^2 trap, and practice 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.
What is the completing the square formula?
x² + bx = (x + b/2)² − (b/2)². In words: half the x-coefficient goes inside the square, and its square is subtracted (or added to the other side) to keep the expression equal.
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 double-checked for accuracy before it reaches you.

Related Algebra 2 skills

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