Today:

Warm-Up: Number SenseWarm-Up: Review Quadratic

GraphsIntroduction to Completing the

SquareClass Work

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Number Sense: Time

It is 12:00 noon on a February Friday in Saipan. What time and

day is it in... 1. Tokyo, Japan

4. Honolulu, Hawaii

3. South Pole, Antarctica

2. New York, NY

1. Friday, 11:00 am

4. Thursday, 4:00 pm

3. Friday, 3:00 pm

2. Friday, 3:00 am

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1. x3 - 121x = 0

2. Using mental math, find the product of (x + 1) 23. Find the dimensions of the

rectangle:

x - 3

x - 5A = 120

Factor completely:

Do not say the answer out loud please

4. 90% of 90 girls and 80% of 110 boys have shown up at the stadium on time. How many people are late?

Warm-Up:

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Warm-Up:

5. If the parabola y = x2 is flipped upside down, made 5 times as wide, and shifted 8 units down the y-axis, write the equation for the new parabola.

6. If the parabola y = x2 is flipped upside down, made twice as narrow, and shifted 6 units up the y-axis, write the equation for the new parabola.

y = -⅕x2 - 8

y = -2x2 + 6

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Class Notes Section of Notebook

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7

Solve by Taking Square Root:

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Solve by Taking Square Root:

5x2 = 20

x2 = 36

5x2 - 45 = 0

x2 = 32 The radical must be simplified. This means all perfect squares must be factored out of the radical

= = 4

Which leads us to our new method for solving quadratics.

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Completing the Square:

Factor completely: 2x2 + 16x - 20 = 0

If we try to factor using only the tools we have now, we will not be able to solve the quadratic.What we need is another tool in our toolbox.

This will allow us to solve quadratic equations that were previously ‘unsolvable’.

This new tool is called Completing the Square

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Completing the SquareFactoring “unfactorable” 2nd degree

trinomials

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• We have learned earlier that a perfect square trinomial can always be factored.

• Therefore, if we have a trinomial we cannot factor using integers, we can change it in such a way that we are dealing with a perfect square trinomial.

Completing the Square:

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• Recall that a perfect square trinomial is always in the form:

• Therefore, we have to change the polynomial so that it fits the form.

• To really learn this, go through each step of the process. Your goal should be to learn the steps in order.

22 2 baba

Completing the Square:

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The equation we are going to solve is the following…

By testing whether or not the factors of c can sum to equal b, we can determine if the trinomial is factorable. This trinomial is not factorable in its present form. However, with our new tool, we can solve this previously 'unsolvable' quadratic.

2

2 16 20 0x x

There are five steps in this process, let's write them down.

Completing the Square:

Step 1Divide by the leading coefficient to set the a-value to 1.

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2

2

2 16 20 02

8 10 0

x x

x x

Completing the Square:

Step 2 Re-write the equation in the form ax + by = c

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2

2

2

8 10 0

8 10 10 10

18 0

0

x x

x x

x x

Completing the Square:

Step 3Find one-half of the b value. Add the square of that number to both sides.

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2

2

2

2 2

8 10

8 104 4

16 28 6

x x

x x

x x

2

2

b

Completing the Square:

Step 4A) Re-write the perfect square trinomial as a binomial squared.

B) Find the square root of each side of the equation.

17

2

24

8 16 26

26

264

x

x

x x

Completing the Square:

Step 5 Solve for x.

18

4 2

4

2

6

264

6

4

4

x

x

x

Completing the Square:

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Completing the Square:Example 2

x2 - 16x +15 = 0

Re-write the equation in the form ax + by = c

Divide by leading coefficient

x2 - 16x = -15

2

2

b

Take one-half of b, then square it. Add the square to both sides.

x2 - 16x + 64 = -15 + 64

Simplify both sides.x2 - 16x + 64 = -

15 + 64

(x - 8) 2 = 49

1) Find the value of 2

2

b

2x bx

Completing the Square:

2. Add the value to the expression, this completes the square

2 6x x

2 10x x

Solving an Equation by Completing the Square

Class Work: See Handout

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