How do I solve a quadratic equation by completing the square?

The key is to make one side a perfect-square trinomial. If the coefficient of x² is 1, move the constant to the other side. Take half the coefficient of x, square that number, and add it to both sides. The left side can then be written as a squared binomial. Take the square root of both sides, including both the positive and negative possibilities, and isolate x. [S1]

Here is a worked example with the transformations labeled:

1. Start: x² + 6x + 2 = 0
2. Move the constant: x² + 6x = -2
3. Complete the square: Half of 6 is 3, and 3² = 9. Add 9 to both sides: x² + 6x + 9 = 7.
4. Write as a square: (x + 3)² = 7
5. Take square roots: x + 3 = ±√7
6. Solve: x = -3 ± √7

The ± matters because both √7 and -√7 square to 7. Completing the square works by adding the same amount to both sides until one side is a perfect square. [S1]

First divide every term in the equation by the nonzero coefficient of x². That makes the x² coefficient 1; then use the same steps. [S1]

For example, solve 2x² + 8x - 3 = 0:

1. Make the x² coefficient 1: Divide the whole equation by 2: x² + 4x - 3/2 = 0.
2. Move the constant: x² + 4x = 3/2.
3. Complete the square: Half of 4 is 2, and 2² = 4. Add 4 to both sides: x² + 4x + 4 = 3/2 + 4 = 11/2.
4. Write as a square: (x + 2)² = 11/2.
5. Take square roots and solve: x + 2 = ±√(11/2), so x = -2 ± √(22)/2.

The radical can also be left as √(11/2). The essential change is dividing through by the leading coefficient before completing the square. [S1]

Check each line carefully: division must apply to every term, and when you add the completing term, add it to both sides. Keep fractions exact while simplifying. After you find the solutions, substitute each one into the original equation to check that it satisfies that equation.

For the first example, the completed-square equation is (x + 3)² = 7. The two values x = -3 + √7 and x = -3 - √7 make x + 3 equal √7 and -√7, respectively; each gives a square of 7. If your work reaches a negative number on the right just before taking a real square root, recheck the transformations and arithmetic rather than proceeding as though it were a positive number.

Use this layout, replacing the bracketed values with your equation’s coefficients:

Equation: ax² + bx + c = 0

1. Normalize: If a ≠ 1, divide every term by a: x² + (b/a)x + c/a = 0.
2. Isolate the x terms: x² + (b/a)x = -c/a.
3. Find the completing term: Half of b/a is b/(2a); square it to get b²/(4a²). Add this same value to both sides.
4. Factor the square: (x + b/(2a))² = -c/a + b²/(4a²).
5. Take square roots: x + b/(2a) = ±√(-c/a + b²/(4a²)).
6. Isolate x and check both values in the original equation.

Keep the sign attached to the constant when moving it, and use parentheses when squaring a negative half-coefficient. The template assumes a nonzero coefficient of x². [S1]