How to Solve by Completing the Square- Tutorial

What Completing the Square Actually Is

Completing the square is a technique that converts any quadratic equation from standard form ax² + bx + c = 0 into vertex form a(x - h)² + k = 0. The name comes from what you're literally doing: creating a perfect square trinomial from a binomial expression.

It's not the fastest method for every problem. Sometimes factoring works faster. Sometimes you just memorize the quadratic formula. But completing the square is the method that actually teaches you why the quadratic formula works, and it's essential when you need to graph parabolas or find maximum/minimum values.

When to Use This Method

You should complete the square when:

• The quadratic doesn't factor cleanly
• You need to convert to vertex form for graphing
• You're deriving the quadratic formula (which is completing the square on ax² + bx + c = 0)
• The problem specifically asks for it
• You need to find the vertex of a parabola

If you just need to solve for x and the equation factors, use factoring. This method exists for situations where factoring fails.

The Step-by-Step Process

Standard Case: a = 1

When your quadratic already has a leading coefficient of 1, the process is straightforward:

1. Move the constant term to the right side
2. Add the square of half the x-coefficient to both sides
3. Factor the left side as a perfect square
4. Solve using square roots

Example 1: x² + 6x + 5 = 0

Step 1: x² + 6x = -5

Step 2: Half of 6 is 3. Square it: 3² = 9. Add 9 to both sides.

x² + 6x + 9 = -5 + 9

Step 3: Factor the left side.

(x + 3)² = 4

Step 4: Take the square root of both sides.

x + 3 = ±2

Step 5: Solve for x.

x = -3 + 2 = -1 or x = -3 - 2 = -5

Verify: (-1)² + 6(-1) + 5 = 1 - 6 + 5 = 0 ✓

When a ≠ 1: Factor Out First

When the leading coefficient isn't 1, you must factor it out of the x-terms before completing the square.

Example 2: 2x² + 8x - 10 = 0

Step 1: Divide everything by 2 (or factor 2 out).

x² + 4x - 5 = 0

Step 2: Move the constant.

x² + 4x = 5

Step 3: Half of 4 is 2. Square it: 2² = 4. Add to both sides.

x² + 4x + 4 = 5 + 4

Step 4: Factor.

(x + 2)² = 9

Step 5: Solve.

x + 2 = ±3

x = 1 or x = -5

The Formula Trick (Shortcut)

Once you've done this enough, you'll notice a pattern. For any equation x² + bx = c, the completed square form is:

(x + b/2)² = c + (b/2)²

You can skip writing out each step and jump straight to the completion. This works for coefficients of 1 only—remember that.

Common Mistakes That Will Kill Your Answer

• Forgetting to balance both sides — whatever you add to the left, you must add to the right
• Squaring half the coefficient wrong — half of 8 is 4, and 4² = 16, not 8
• Not factoring out a first when a ≠ 1 — this is the most common error
• Losing the negative sign — (x - h)² means h is subtracted, not added
• Dropping the ± symbol — square roots have two solutions

Completing the Square vs. Other Methods

Most math classes assign completing the square because it builds understanding, not because it's always the fastest way to get an answer.

Converting to Vertex Form

The real power of completing the square is converting to vertex form f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola.

Example: Convert y = 2x² + 12x + 7 to vertex form

Step 1: Factor out the coefficient of x² from the first two terms.

y = 2(x² + 6x) + 7

Step 2: Complete the square inside the parentheses. Half of 6 is 3, and 3² = 9.

y = 2(x² + 6x + 9 - 9) + 7

Step 3: Factor the perfect square and distribute the -9.

y = 2[(x + 3)² - 9] + 7

y = 2(x + 3)² - 18 + 7

Step 4: Simplify.

y = 2(x + 3)² - 11

The vertex is (-3, -11). The parabola opens upward (a = 2 > 0) and is narrower than y = x².

Practice Problems

Try these without looking at the answers first:

1. x² + 4x - 12 = 0
2. x² - 10x + 24 = 0
3. 3x² + 6x - 9 = 0
4. Convert y = x² - 8x + 3 to vertex form

Answers: 1) x = 2 or x = -6 | 2) x = 4 or x = 6 | 3) x = 1 or x = -3 | 4) y = (x - 4)² - 13

The Bottom Line

Completing the square isn't magic—it's arithmetic. Take half of the x-coefficient, square it, add it to both sides, and factor. The process is mechanical once you internalize the steps. Practice 10 problems and you'll have it down.