GOZUNTA
Try it now
All courses Algebra I · A-SSE.3.b 58 of 76
Complete the square to reveal maximum or minimum values of quadratic functions.

Complete the square for a monic quadratic

Problem
Rewrite \(x^{2}+8x+3\) in vertex form by completing the square. Show the half-coefficient and square calculation, compensate for the added value, and identify the vertex.
Your answer
Show answer choicesHide answer choices Work through the mathematics first, then compare your reasoning.Select the answer that matches your work.
With a free account

Practice the problem type you meant to practice.

Navigate by objective and problem type instead of taking whichever free problem happens to come next.

Create a free account
With paid access

Build sessions from the full problem bank.

Choose the objective you need and make a learning session using all of its available problem types and variants.

Compare plans

Hint

Compute b/2 and(b/2)² exactly.

x²+bx=(x+b/2)²−(b/2)². Preserve equivalence by adding and subtracting the same square.

Solution walkthrough

01

Take half the x-coefficient

\[\begin{aligned} x^2+8x+3 \\ 8/2=4 \end{aligned}\]

For a monic quadratic, the binomial constant used to complete the square is half the signed linear coefficient.

02

Square and compensate

\[\begin{aligned} 4^2=16 \\ x^2+8x+16-16+3 \end{aligned}\]

Adding 16 creates a perfect-square trinomial. Subtracting the same 16 keeps the expression equivalent, and the original constant 3 remains.

03

Write vertex form

\[\begin{aligned} x^2+8x+16=(x+4)^2 \\ (x+4)^2-16+3=(x+4)^2-13 \end{aligned}\]

The completed trinomial is x plus 4 squared, and the outside constants combine to -13.

04

Identify and verify the vertex

\[\begin{aligned} \text{vertex}=(-4,-13) \\ (x+4)^2-13=x^2+8x+16-13=x^2+8x+3 \end{aligned}\]

Vertex form is (x-h)²+k. Since x+4=x-(-4), h=-4 and k=-13. Expansion returns the original expression.

+

Another way

  1. Use h=-b/(2a)=-4 and evaluate the original at -4 to get k=-13.

!

Common mistake

Do not add and subtract only the half-coefficient 4. The completing value is its square, 16.

Solution walkthrough video