3.5 - Completing the Square

3.5 - Completing the Square

Math II

Unit 3 - Quadratic Functions

Mr. Coleman

Learning Targets:

• I can identify a quadratic in the form of a perfect square trinomial.

• I can rewrite quadratic functions into perfect square trinomial form by "completing the square."

• I can solve quadratic functions using the "completing the square" technique.

Perfect Square Trinomials

Quadratic equations that can be rewritten in the form of

 $\left(x+a\right)^2$   or  $\left(x-a\right)^{2\ }$  

Think About It:

What value should go in the blank to complete this square - and what would be the perfect square trinomial we get as a result?

(Hint - don't overthink it, use the area model - what are the dimensions of the purple square?)

Multiple Choice

Based on this area model, which of the following perfect square trinomials should be equal to

 $\left(x+1\right)^2$  ?

•  $x^2+2x+1$  

•  $x^2+x+1$  

•  $x^2+2x+2$  

Open Ended

Draw out an area model like the previous example to determine the perfect square trinomial for

 $\left(x-2\right)^2$  .

The Pattern

The expanded form  $\left(x+a\right)^2$   of will have a middle coefficient that is DOUBLE the value of a, and a constant that is the SQUARE of a.

(Note the sign change for  $\left(x-a\right)^2$ )

Multiple Choice

Which of the following is equivalent to

 $x^2+8x+16$  ?

•  $\left(x+8\right)^2$  

•  $\left(x-4\right)^2$  

•  $\left(x+4\right)^2$  

•  $\left(x-8\right)^2$  

• ...wut?

Completing the Square

• An algebraic process where we "force" a quadratic equation into a perfect square trinomial.

• Useful as a means of solving quadratics - once we can rewrite as

   $\left(x\pm a\right)^2$  , we can then solve by extracting square roots - no more need to factor! 

• The algorithm can be tricky at first, but with practice you'll get it!

The Process

1) Move c term to the other side.

2) Take HALF of the b value, SQUARE it, add to BOTH sides.

3) Rewrite left side as binomial square.

4) Solve by extracting square roots.

Open Ended

What is the first step to solve $x^2-8x-9=0$  by completing the square?

Multiple Choice

Which of the following values would you then add to both sides in the next step of completing the square?

• Add 8 to both sides

• Add 64 to both sides

• Add 16 to both sides

• Add 4 to both sides

Multiple Choice

Which of the following equations would be obtained in the next step of completing the square?

• $\left(x+8\right)^2=25$

• $\left(x-4\right)^2=25$

• $\left(x-8\right)^2=25$

• $\left(x-16\right)^2=25$

Open Ended

What are the final values of x you obtain as solutions in your final step?

Open Ended

Solve by completing the square: $6x^2+60x+582\ =\ 0$  

(HINT: divide out by GCF first to bring the values down!)