How to Complete the Square

Step-by-step solution

1. Move the constant to the right side.

   $$x^2 + 6x = -5$$

   Why: The square is easier to build when the left side holds only x-terms.

2. Take half of 6 and square it.

   $$\left(\frac{6}{2}\right)^2 = 9$$

   Why: The number that completes the square is half the middle coefficient, squared.

3. Add 9 to both sides.

   $$x^2 + 6x + 9 = 4$$

   Why: Adding the same number to both sides keeps the equation balanced.

4. Write the left side as a square.

   $$(x + 3)^2 = 4$$

   Why: The left side is now a perfect square trinomial: x^2 + 6x + 9 = (x + 3)^2.

5. Take square roots and solve both cases.

   $$x + 3 = \pm 2 \Rightarrow x = -1,\ x = -5$$

   Why: A squared expression has two square roots, positive and negative.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

1. 1

   $x^2 + 4x + 3 = 0$

   Show answer

   $x = -1,\ x = -3$

2. 2

   $x^2 + 8x + 12 = 0$

   Show answer

   $x = -2,\ x = -6$

3. 3

   $x^2 - 10x + 21 = 0$

   Show answer

   $x = 3,\ x = 7$

4. 4

   $x^2 + 2x - 8 = 0$

   Show answer

   $x = 2,\ x = -4$

5. 5

   $x^2 - 6x + 8 = 0$

   Show answer

   $x = 2,\ x = 4$