Complete the square

Key Notes :

Understanding Completing the Square

  • Completing the square is a technique used to rewrite a quadratic equation in the form (x + h)² = k, where h and k are constants.
  • This form makes it easier to solve for x by taking the square root of both sides.

Steps to Complete the Square

  1. Ensure the Leading Coefficient is 1: If the leading coefficient (a) is not 1, divide the entire equation by a to make it 1.
  2. Isolate the Quadratic and Linear Terms: Move the constant term (c) to the other side of the equation.
  3. Add the Square of Half the Linear Coefficient: Add (b/2)² to both sides of the equation. This will create a perfect square trinomial on the left side.
  4. Factor the Perfect Square Trinomial: The left side of the equation should now be in the form (x + h)².
  5. Solve for x: Take the square root of both sides and solve for x.

Example

Solve: x² – 6x + 2 = 0

  • Step 1: The leading coefficient is already 1.
  • Step 2: Move the constant term: x² – 6x = -2
  • Step 3: Add the square of half the linear coefficient: x² – 6x + 9 = -2 + 9
  • Step 4: Factor the perfect square trinomial: (x – 3)² = 7
  • Step 5: Take the square root: x – 3 = ±√7
    • x = 3 ± √7

Key Points

  • Completing the square is a useful technique for solving quadratic equations, especially when factoring is not straightforward.
  • The goal is to create a perfect square trinomial on the left side of the equation.
  • Adding the square of half the linear coefficient is the key step in completing the square.
  • Once the equation is in the form (x + h)² = k, solving for x is straightforward by taking the square root.

Learn with an example

Complete the square. Fill in the number that makes the polynomial a perfect-square quadratic.

u2 − 16u +________

Add (b/2)2 to complete the square.

u2-16u+(b/2)2

=u2 -16u+(-16/2)2 Plug in b =-16

=u2 -16u+(-8)2 Divide

=u2 -16u+64 Square

This quadratic can be written as a square,(u − 8)2,so it is a perfect-square quadratic. The number needed to complete the square was 64.

Complete the square. Fill in the number that makes the polynomial a perfect-square quadratic.

k2 − 6k +________

Add (b/2)2 to complete the square.

k2-6k+(b/2)2

=k2 -6k+(-6/2)2 Plug in b = -6

=k2 -6k+(-3)2 Divide

=k2 -6k+9 Square

This quadratic can be written as a square,(k − 3)2,so it is a perfect-square quadratic. The number needed to complete the square was 9.

Complete the square. Fill in the number that makes the polynomial a perfect-square quadratic.

f2 + 20f +________

Add (b/2)2 to complete the square.

f2+20f+(b/2)2

=f2 +20f+(20/2)2 Plug in b = 20

=f2 +20f+(10)2 Divide

=f2 +20f+100 Square

This quadratic can be written as a square,(f + 10)2,so it is a perfect-square quadratic. The number needed to complete the square was 100.

let’s practice!