LEMMA

Polynomials & quadratics

Complete the square

ALEKS placement

Add the square of half the linear coefficient to build a perfect-square trinomial.

What this covers

x2+bx+(2b​)2=(x+2b​)2
  • Divide through so the leading coefficient is 1.
  • Move the constant to the other side.
  • Halve the x-coefficient, square it, and add it to both sides.
  • Why half of b, squared: the two numbers that multiply to (b/2)² and add to b are b/2 and b/2, so the trinomial is a perfect square.
  • Write the left side as a squared binomial: x, the sign of the x-term, then the half you had before squaring.
  • Take square roots of both sides and keep both signs.

Use this when the question asks for vertex form, the centre of a circle, or specifically says to complete the square. For roots, try the factoring method you already know first. If it will not factor and the x-coefficient is even, completing the square keeps the numbers small; otherwise the next quadratic method handles it.

Worked example

Worked example

Solve x² + 6x − 7 = 0 by completing the square

  1. x2+6x=7Move the constant across.
  2. (6÷2)2=9Halve the linear coefficient, then square that half. This is the missing value that makes a squared group.
  3. x2+6x+9=7+9Add the missing value to both sides so the equation stays balanced.
  4. (x+3)2=16The left side is now a perfect square.
  5. x+3=4orx+3=−4Positive four and negative four both square to sixteen. Each gives a possible solution.
  6. x=1orx=−7Subtract three in each case. Both values satisfy the original equation.
Another worked example

Rewrite x² − 6x + 11 in vertex form.

  1. x2−6x+9−9+11Half of −6 is −3, and (−3)² = 9. Adding 9 and subtracting 9 adds zero, so the value is unchanged.
  2. (x−3)2+2x² − 6x + 9 is the square (x − 3)². The leftover numbers give −9 + 11 = 2.

A common mistake

A mistake Lemma catches

From x2−4x=5

x2−4x+4=5x2−4x+4=5+4

“Adding (−4 ÷ 2)² = 4 completes the square on the left, so 4 has to be added to the right as well: 5 + 4 = 9. Adding it to one side only changes the equation.”

Try one

Sample problem

Rewrite as a squared binomial plus or minus a constant.

x2+12x

Practice complete square free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

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