Quadratic Equations • Topic 4 of 5

Completing the Square

Completing the square rewrites x² + bx + c as a perfect square plus a constant, which then solves by square roots and reveals the vertex. Take half of b, square it, and add and subtract it: x² + 6x = (x + 3)² − 9. To solve x² + 6x + 5 = 0, write (x + 3)² − 9 + 5 = 0, so (x + 3)² = 4 and x = −1 or −5. The method works for any quadratic (divide by a first if a ≠ 1) and is the source of the quadratic formula. The SAT uses it to convert to vertex form and to solve when factoring is awkward.

✅ Solved examples

1. Complete the square for x² + 6x.
Half of 6 is 3; 3² = 9, so x² + 6x = (x + 3)² − 9.
2. Solve x² + 4x − 5 = 0 by completing the square.
x² + 4x = (x + 2)² − 4, so (x + 2)² − 4 − 5 = 0 → (x + 2)² = 9 → x = 1 or −5.
3. Write x² − 2x + 1 as a perfect square.
It is already (x − 1)².
4. Complete the square for x² − 8x.
Half of −8 is −4; (−4)² = 16, so x² − 8x = (x − 4)² − 16.

✏️ Practice — try these, take hints as needed

1. Complete the square for x² + 10x.
Half of 10 is 5.
5² = 25.
(x + 5)² − 25.
(x + 5)² − 25.
2. Complete the square for x² − 6x.
Half of −6 is −3.
(−3)² = 9.
(x − 3)² − 9.
(x − 3)² − 9.
3. Solve x² + 2x − 3 = 0 by completing the square.
x² + 2x = (x + 1)² − 1.
(x + 1)² − 1 − 3 = 0 → (x + 1)² = 4.
x + 1 = ±2.
x = 1 or x = −3.
4. Write x² + 8x + 16 as a perfect square.
Half of 8 is 4, 4² = 16.
Matches the constant.
—
(x + 4)².
5. Complete the square for x² + 12x.
Half of 12 is 6.
6² = 36.
(x + 6)² − 36.
(x + 6)² − 36.

📝 Topic test — 8 questions

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