Completing the Square
In short: Completing the square rewrites x² + bx as a perfect square by adding (b/2)², which turns a hard quadratic into one you can solve with a square root.
Algebra 1AlgebraA-REI.B.4a
What is Completing the Square?
Imagine \(x^2 + 6x\) as an \(x\) by \(x\) square plus two strips of width 3, with a corner of area 9 missing, and completing the square supplies that piece so the shape becomes \((x + 3)^2\).
The rule: for \(x^2 + bx\), add \(\left(\frac{b}{2}\right)^2\). When solving an equation, add the same amount to both sides. The result takes the form \((x + p)^2 = q\), which you finish by taking square roots, remembering both signs.
How to spot it
- A quadratic does not factor with whole numbers, yet exact answers with radicals are wanted.
- The directions say rewrite in vertex form, or solve by completing the square.
- The coefficient of x² is 1 and the x term has an even number in front, which makes halving easy.
- A question asks what must be added to \(x^2 + 14x\) to make a perfect square.
The method
- Move the constant to the right side so only \(x^2 + bx\) stays on the left.
- Halve b, square it, and add that number to both sides.
- Rewrite the left side as \((x + \frac{b}{2})^2\) and simplify the right side.
- Take the square root of both sides, using plus and minus, then solve for x.
Completing the Square: solved example
Problem
Solve \(x^2 – 10x + 7 = 0\) by completing the square, and give both exact and decimal answers.
Solution
Move 7: \(x^2 – 10x = -7\). Half of \(-10\) is \(-5\), and its square is 25. Add 25 to both sides: \(x^2 – 10x + 25 = 18\).
That is \((x – 5)^2 = 18\), so \(x – 5 = \pm\sqrt{18} = \pm 3\sqrt{2}\). The solutions are \(x = 5 \pm 3\sqrt{2}\), about 9.24 and 0.76.
Try it
Question 1
What number completes the square in \(x^2 + 14x + \_\_\)?
Show the answer
\(\left(\frac{14}{2}\right)^2 = 49\).
Question 2
Solve \(x^2 + 8x = 20\).
Show the answer
Add 16: \((x + 4)^2 = 36\), so \(x + 4 = \pm 6\), giving \(x = 2\) or \(x = -10\).
Completing the Square vs. Quadratic Formula
Both solve any quadratic, and the formula is simply this process carried out once on the general equation. Completing the square shines when you need vertex form or an exact answer by hand with a leading coefficient of 1; the formula is quicker when the numbers are awkward.
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