Completing the square

Algebra 2 Tutorial

Completing the square

Intro

Completing the square is a technique that changes the form in order to easily find the maximum or minimum of the quadratic expression.

Sample Problem

Complete the square for the following quadratic equation

x^2+8x-4=0

Solution

How to complete the square:
1) Move the constant, often referred to as the “c” term. To the other side of the equal sign.
2) Use the coefficient of the x-term. Divide it by two and then square it. This quantity gets added to both sides of the equation.
3) Factor the expression. Hint: the digit is the number you got when you divided the coeffient by two.
4) Set the equation to zero by moving the constant back to the other side of the equal sign.
Negative leading coefficient: Factor out the negative and then proceed as above. Be sure to include the negative in the final answer.
Example:
x^2+8x-4=0
x-term is 8x
constant is -4

x^2+8x-4=0 Original equation
x^2+8x=4 Move the constant by adding it to both sides.
8÷2=4 ; 4^2=16 Divide the coefficient and then square it.
x^2+8x+16=20 Add the result to both sides of the equation
〖(x+4)〗^2=20 Factor the expression. Notice the 4 is the result when dividing
the coefficient by two.

〖(x+4)〗^2-20=0 Set the equation to zero by subtracting the constant from both
sides.



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