Quadratic to Vertex

Quadratic Form to Vertex form

AKa: Completing the square to rename Quadratics​

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By Eniola Ajayi

​Learning Targets

• ​Identify a quadratic in form of a perfect square trinomial

• Rewrite quadratic function in standard form into perfect square trinomial/vertex form by "completing the square

• ​Solve quadratic functions using the "completing the square" technique

Subject | Subject

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Make sure you take notes as you go through the slides

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perfect Square Trinomials

​Quadratic equations that can be rewritten in the form of

(x+a)² or (x-a)²

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

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HINT: Don't overthink it, use the area model - what are the dimension of the purple square?​

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​think about it

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Multiple Choice

Based on this area model, which of the following perfect square trinomials should be equal to (x +1)²?

• x²+2x +1

• x² + x +1

• x² +2x+1

Open Ended

Draw out an area model like the previous example to determine the perfect square trinomial for (x -2)²

Open Ended

PERFECT SQUARES: Distribute/FOIL – write all four terms out!

(x+3)²

Open Ended

PERFECT SQUARES: Distribute/FOIL – write all four terms out!

(2x -1)²

Open Ended

What do you notice about second and third terms?

The expanded form of (x + a)² will have a middle coefficient that is DOUBLE the value of a, and a constant that is the square of a.

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(Note the sign change for (x - a)²

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The Pattern

Multiple Choice

Which of the following is equivalent to

x² + 8x + 16?

• (x - 8)²

• (x+8)²

• (x +4)²

• (x- 4)²

• ...wut?

​What is completing the square?

• An algebraic process where we go from standard form to vert​ex form.

• Useful as as way to solve quadratics in standard form without factoring.

• It will take practice to get comfortable with the process, but do not give up!​

Subject | Subject

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1. Move the c term to the other side.

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

3. Rewirte left side as binomial square.

4. Solve by using the square root method​

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​The process

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Open Ended

What is the first step to solve x² - 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 4 to both sides

• Add 16 to both sides

• Add 64 to both sides

Multiple Choice

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

• (x - 16)² = 25

• (x -8)² = 25

• (x+8)² = 25

• (x - 4)² =25

Open Ended

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

Multiple Select

Exit Ticket A: Select ALL that are perfect squares

• (x+5)(5+x)

• (x+5)(x-5)

• (x-3)²

• x²+8x+16

• x-3²

Fill in the Blank

Exit Ticket B: What number must be added to make this a perfect square trinomial?

x² - 12x + ____ (just type the number)

Multiple Choice

Exit Ticket C:

Complete the Square x² + 10x = -13 and then write the equation in vertex form.

• (x +10)² = 13

• (x- 5)² + 8

• (x +5)² - 12

• (x - 100)² - 83