Algebra

Complete the square

Rearrange pieces. Find the turning point.

Predict, test and explain

Step 1 of 3

Predict

Completing the square for x² + 4x + 1 gives (x + 2)² plus which correction?

x2+4x+1x^2 + 4x + 1
x2x^2
b2x\frac{b}{2}x
b2x\frac{b}{2}x
xx
xx
b2\frac b2
b2\frac b2

Scroll the diagram sideways to see every label.

b2=2\frac b2 = 2constant=1\text{constant} = 1

InvestigateWhat must you add to fill the missing corner?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Complete the square.

x2+6x+5x^2+6x+5

Hint

Halve the coefficient of the linear term, then square it.

Reveal answer

The completed square adds nine, so subtract four to keep the original constant.

x2+6x+5=(x+3)2−4x^2+6x+5=(x+3)^2-4

Build an explanation

Hint 1

Split the x-term equally between two strips.

Hint 2

Half the coefficient of x gives the side of the missing corner.

Hint 3

Add that corner to make a square, then subtract the same amount to keep the expression unchanged.

Worked example

This example uses fixed values, separate from the diagram controls.

Complete the square for x² + 4x + 1.

  1. Half of 4 is 2; the missing corner has area 2² = 4.
  2. x² + 4x + 1 = x² + 4x + 4 − 3.
  3. The first three terms form (x + 2)².

Answer: (x + 2)² − 3.

Connect this idea

Watch out: Adding a corner changes the value unless you subtract the same area.