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SAT Problem of the Day

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5.3 Factoring Quadratic Expressions 5.3 Factoring Quadratic Expressions Objectives: Factor a quadratic expression Use factoring to solve a quadratic equation and find the zeros of a quadratic function

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Page 1: SAT Problem of the Day. 5.3 Factoring Quadratic Expressions 5.3 Factoring Quadratic Expressions Objectives: Factor a quadratic expression Use factoring

SAT Problem of the Day

Page 2: SAT Problem of the Day. 5.3 Factoring Quadratic Expressions 5.3 Factoring Quadratic Expressions Objectives: Factor a quadratic expression Use factoring

SAT Problem of the Day

Page 3: SAT Problem of the Day. 5.3 Factoring Quadratic Expressions 5.3 Factoring Quadratic Expressions Objectives: Factor a quadratic expression Use factoring

5.3 Factoring Quadratic Expressions5.3 Factoring Quadratic ExpressionsObjectives: •Factor a quadratic expression•Use factoring to solve a quadratic equation and find the zeros of a quadratic function

Page 4: SAT Problem of the Day. 5.3 Factoring Quadratic Expressions 5.3 Factoring Quadratic Expressions Objectives: Factor a quadratic expression Use factoring

Example 1Factor each quadratic expression.

a) 27x2 – 18x9

factor out the GCF for all terms(3x – 2)

27 18x2 xx

b) 5x(2x + 1) – 2(2x + 1)

factor out the GCF for all terms

(2x + 1)(2x + 1)

( )5x - 2

(2x + 1)

Factor a quadratic expression

Page 5: SAT Problem of the Day. 5.3 Factoring Quadratic Expressions 5.3 Factoring Quadratic Expressions Objectives: Factor a quadratic expression Use factoring

Factoring x2 + bx + cTo factor an expression of the form ax2 + bx + c, where a = 1, look for integers r and s such that r s = c and r + s = b. Then factor the expression.

x2 + bx + c =

(x + r)(x + s)

Factor a quadratic expression

Page 6: SAT Problem of the Day. 5.3 Factoring Quadratic Expressions 5.3 Factoring Quadratic Expressions Objectives: Factor a quadratic expression Use factoring

Example 2Factor x2 + 12x + 27.

( ) ( ) x x +

+

3 9

Factor a quadratic expression

Page 7: SAT Problem of the Day. 5.3 Factoring Quadratic Expressions 5.3 Factoring Quadratic Expressions Objectives: Factor a quadratic expression Use factoring

Example 3Factor x2 - 15x - 54.

( ) ( )

x x +

- 3 18

Factor a quadratic expression

Page 8: SAT Problem of the Day. 5.3 Factoring Quadratic Expressions 5.3 Factoring Quadratic Expressions Objectives: Factor a quadratic expression Use factoring

Example 4Factor 5x2 + 14x + 8.

( ) ( ) 5x x +

+

4 2

Factor a quadratic expression

Page 9: SAT Problem of the Day. 5.3 Factoring Quadratic Expressions 5.3 Factoring Quadratic Expressions Objectives: Factor a quadratic expression Use factoring

Practice

1) 5x2 + 15x Factor.

2) (2x – 1)4 + (2x – 1)x

3) x2 + 9x + 20 4) x2 – 7x - 30

5) 3x2 + 11x - 20

Factor a quadratic expression

Page 10: SAT Problem of the Day. 5.3 Factoring Quadratic Expressions 5.3 Factoring Quadratic Expressions Objectives: Factor a quadratic expression Use factoring

Special ProductsFactoring the Difference of Two Squares

a2 – b2 = (a + b)(a – b)

Factoring Perfect-Square Trinomials

a2 + 2ab + b2 = (a + b)(a + b)a2 - 2ab + b2 = (a - b)(a - b)

Factor a quadratic expression

Page 11: SAT Problem of the Day. 5.3 Factoring Quadratic Expressions 5.3 Factoring Quadratic Expressions Objectives: Factor a quadratic expression Use factoring

Example 5Factor x2 - 16.

( ) ( ) x x +

- 4 4

Factor a quadratic expression

Page 12: SAT Problem of the Day. 5.3 Factoring Quadratic Expressions 5.3 Factoring Quadratic Expressions Objectives: Factor a quadratic expression Use factoring

x2

Example 6Factor x4 - 81.

( ) ( )

x2 +

- 9 9 ( )

(x2 + 9)

( )

x x +

- 3 3

Factor a quadratic expression

Page 13: SAT Problem of the Day. 5.3 Factoring Quadratic Expressions 5.3 Factoring Quadratic Expressions Objectives: Factor a quadratic expression Use factoring

x - 6

Example 7Factor 2x2 – 24x + 72.2 2

4 72

2 ( ) x2 – 12x + 36 2( )( ) x - 6

Factor a quadratic expression

Page 14: SAT Problem of the Day. 5.3 Factoring Quadratic Expressions 5.3 Factoring Quadratic Expressions Objectives: Factor a quadratic expression Use factoring

Zero-Product Property

If pq = 0, then p = 0 or q = 0.

Find the zeros of a quadratic function

Page 15: SAT Problem of the Day. 5.3 Factoring Quadratic Expressions 5.3 Factoring Quadratic Expressions Objectives: Factor a quadratic expression Use factoring

Example 8Solve .5x2 + 7x = 0

x(5x + 7) = 0x = 0 5x + 7 = 0or

5x = -77x 5

CHECK: 5x2 + 7x = 05(0)2 + 7(0) = 0

0 + 0 = 0

CHECK: 5x2 + 7x = 027 75 7 05 5

49 495 025 5

Find the zeros of a quadratic function

Page 16: SAT Problem of the Day. 5.3 Factoring Quadratic Expressions 5.3 Factoring Quadratic Expressions Objectives: Factor a quadratic expression Use factoring

Example 9Find the zeroes of the functionf(x) = x2 – 5x + 6

x2 – 5x + 6 = 0

x -3 = 0 x - 2 = 0orx = 2

CHECK: x2 – 4x = x - 632 – 4(3) = 3 - 6

9 – 12 = -3

CHECK: x2 – 4x = x - 6

(x – 3)(x – 2) = 0

x = 3

-3 = -3

22 – 4(2) = 2 - 64 – 8 = -4

-4 = -4

The zeroes are located at x = 2 and x = 3.Find the zeros of a quadratic function

Page 17: SAT Problem of the Day. 5.3 Factoring Quadratic Expressions 5.3 Factoring Quadratic Expressions Objectives: Factor a quadratic expression Use factoring

PracticeFind the zeroes of each function.

1) h(x) = 3x2 + 12x

2) j(x) = x2 + 4x - 21

Find the zeros of a quadratic function

Page 18: SAT Problem of the Day. 5.3 Factoring Quadratic Expressions 5.3 Factoring Quadratic Expressions Objectives: Factor a quadratic expression Use factoring

Collins Type 2

What do you know about the factors of x2+bx+c when c is positive? When c is negative? What information does the sign of b give you in each case?

Factor a quadratic expression

Page 19: SAT Problem of the Day. 5.3 Factoring Quadratic Expressions 5.3 Factoring Quadratic Expressions Objectives: Factor a quadratic expression Use factoring

Homework

Do the problems listed on page 5 of today's packet ("Finding zeros of quadratic functions using factoring"). Some of these are exercises from Lesson 5.3 of the textbook.

Find the zeros of a quadratic functionFactor a quadratic expression