Mr. Scandrett's ClassroomOS

Mathematics · Lesson 09

Completing the Square

Quadratics Perfect squares Vertex form Quadratic formula foundation

Completing the square turns a quadratic into a perfect square so it becomes easier to solve. The whole move comes from one identity: make the expression match (x + p)2 = x2 + 2px + p2.

Learning Target

I can solve a quadratic equation by completing the square.

Students should be able to explain why the added term is (b/2)2, keep both sides balanced, and connect the finished square to vertex form.

Core Idea

The goal is to force the left side into a perfect square. If the expression starts as x2 + bx, then half of b becomes the side extension, and its square becomes the missing corner.

x2 + bx + (b/2)2 = (x + b/2)2

What Is Happening Geometrically?

Imagine x2 as a square. Split the bx rectangle into two equal strips, each with width b/2. Those strips almost make a bigger square, but one corner is missing.

The missing cornerThat corner has side length b/2, so its area is (b/2)2. Add it to both sides to keep the equation balanced.
x2
bx/2
bx/2
(b/2)2

Standard Steps

Start with ax2 + bx + c = 0. If a is not 1, divide every term by a first.

1
Make the leading coefficient 1Divide through by a when the equation starts with ax2.
2
Move the constantRewrite as x2 + bx = -c.
3
Complete the squareTake half of b, square it, and add that value to both sides.
4
Factor the left sideIt becomes (x + b/2)2.
5
SolveTake the square root of both sides, then isolate x.

Worked Example

Solve x2 + 4x + 1 = 0.

x2 + 4x = -1
Move the constant to the right side.
(4/2)2 = 22 = 4
Take half of 4, then square it.
x2 + 4x + 4 = -1 + 4
Add 4 to both sides so the equation stays balanced.
(x + 2)2 = 3
The left side is now a perfect square.
x + 2 = ±√3
Take the square root of both sides.
x = -2 ± √3
Subtract 2 to isolate x.

Interactive Check

Choose the value that must be added to both sides to complete the square.

x2 + 6x = 5
Take half of the x coefficient, then square it.
0/0correct this round

Guided Practice

Type the missing number that completes the square.

x2 + 8x + ___ = (x + 4)2
x2 - 10x + ___ = (x - 5)2
x2 + 6x + ___ = (x + 3)2
x2 - 2x + ___ = (x - 1)2

Why It Matters

Completing the square is more than a trick. It gives vertex form for graphing, explains where the quadratic formula comes from, and still works when a quadratic does not factor cleanly.

Exit Ticket

Students answer these with shown work.

  • Solve x2 + 2x - 8 = 0 by completing the square.
  • Explain why the number added to both sides is based on half of b.
  • How does (x + 2)2 = 3 reveal the vertex of x2 + 4x + 1?

References