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9.5: COMPLETING THE SQUARE:PowerPoint Presentation

9.5: COMPLETING THE SQUARE:

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9.5: COMPLETING THE SQUARE:

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Factoring: A process used to break down any polynomial into simpler polynomials.

9.5: COMPLETING THE SQUARE:

Zero-Product Property: For any real numbers a and b, If ab= 0 Then a= 0 or b = 0.

Complete the square: A process where x2+bx can be changed into a perfect-square trinomial by adding

Procedure:

1) Find the values of a and b.

COMPLETE THE SQUARE: ax2 + bx

2) Use the formula of and add it to ax2+ bx.

3) Re-write ax2+ bx+ as a product of perfect squares.

GOAL:

COMPLETE THE SQUARE:

Ex: What integer is needed to

complete the square:

x2+5x+ ___?

SOLUTION:

Following the steps given

1) a = 1 and b = 5

2) (2.5)

x2+5x+ (2.5)

3) x2+5x+6.25 (x+2.5)(x+2.5)

(x+2.5)

FINDING THE SOLUTION:

(x+2.5)= 0

Using the Zero-Product property

(x+2.5)= 0

(x+2.5) = 0

X = -2.5

The solutions are x = -2.5 twice.

If something happens twicewe only touch the x-axis at that point.

COMPLETING THE SQUARE: ax2 + bx = C

Ex: What is the solutions of the equation:

x2+6X=216

SOLUTION:

x2+6X=216

1) a = 1 and b = 6

2) (3)

x2+6x+(3)=216+(3)

3) x2+6x+9 (x+3)(x+3)

(x+3)= 225

(x+3)=

x = -3 x = -8, 2.

YOU TRY IT:

Ex: What are the solutions of:

r2-4r=30

SOLUTION:

R2-4r=30

1) a = 1 and b = -4

2) (-2)

x2-4x+(-2)=30+(-2)

3) x2-4x+4 (x-2)(x-2)

(x-2)= 34

(x-2)=

x = 2 x = -3.8 and 7.8

REAL-WORLD:

You are planting

A flower garden

Consisting of 3

Square plots

Surrounded by

1 ft border. The total area of the garden and the border is 100tf2.

What is the side length x of each square plot?

SOLUTION:

Adding an x to both sides of the picture we get:

x +2

3x +2

A = b h

A = (x +2)(3x +2)

100= (x +2)(3x +2)

FOIL

100= 3x2+ 8x + 4

To complete the square we now put it in the ax2+bx = c form:

SOLUTION:

100= 3x2+ 8x + 4

3x2+ 8x = 100 - 4

3x2+ 8x = 96

We always prefer a to be 1, so divide by 3:

x2+ x =

SOLUTION: Continue

1) x2+ x =

Now a = 1 , b =

2)

3) x2+ x+ = +

(x + )(x+ )= +

VIDEOS:

SOLVING

BY

FACTORING

Solving by factoring:

http://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/Example%201:%20Solving%20a%20quadratic%20equation%20by%20factoring

VIDEOS:

SOLVING

BY

FACTORING

Solving by factoring:

http://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/Example%202:%20Solving%20a%20quadratic%20equation%20by%20factoring

http://www.khanacademy.org/math/trigonometry/polynomial_and_rational/quad_formula_tutorial/v/solving-quadratic-equations-by-completing-the-square

CLASSWORK:Page 508-509:

Problems: 4, 7, 15, 21, 25, 36, 37.