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All courses Math II · A-REI.4.a 5 of 73
Complete the square to transform quadratics and derive the quadratic formula.

Complete the square for a quadratic of the form x^2 + bx + c

Problem
Complete the square for x²+8x+5.
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Hint

Take half of the \(x\) coefficient and square it. For \(x^2+8x+5\), half of \(8\) is \(4\), and \(4^2=16\).

Completing the square rewrites \(x^2+bx\) as \((x+b/2)^2\) after adding and subtracting the same square value.

Solution walkthrough

01

Find the completing-square value

\[8/2~=~4,~4^2~=~16\]

Half of the linear coefficient is \(4\), and squaring it gives the value needed to complete the square.

02

Add and subtract the same value

\[x^2+8x+5~=~x^2+8x+16-16+5\]

Adding and subtracting \(16\) keeps the expression equal while creating a perfect-square trinomial.

03

Factor the perfect square

\[(x+4)^2-11\]

\(x^2+8x+16\) factors to \((x+4)^2\), and \(-16+5=-11\).

04

State the vertex form

\[(x+4)^2-11\]

This is the completed-square form of the quadratic.

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Another way

  1. You can check by expanding: \((x+4)^2-11=x^2+8x+16-11=x^2+8x+5\).

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Common mistake

A common mistake is to add \(16\) but forget to subtract \(16\), which changes the value of the original expression.