No zeros:

Holt Algebra 2

5-3 Solving Quadratic Equations by Graphing and Factoring

A zero (root) of a function is the x-intercept of the graph. Quadratic functions can have 0, 1, or 2 zeros. (In general, a function can have as many zeros as its highest exponent.)

The zeros of a quadratic function are always symmetric about the axis of symmetry.

Zeroes can be found by graphing or by factoring.No zeros: 1 zero: 2 zeros:

Holt Algebra 2


Factoring by GCF

The GCF (Greatest Common Factor) is the greatest number and/or variable that evenly divides into each term.

Factor each expression by GCF:

1) 4xy2 – 3x 2) 10x2y3 – 20xy2 – 5xy

3) 3n4 + 6m2n3 – 12nm 4) 5x2 + 7

x(4y – 3x) 5xy(2xy2 – 4y – 1)

3n(n3 + 2m2n2 – 6m) Prime

Holt Algebra 2


Determine the zeros of each function:

1)f(x) = 5x2 + 10x

2)g(x) = ½x2 – 2x

3)h(x) = 9x2 + 3x

5x(x + 2)5x = 0 and x + 2 = 0x = 0 and x = -2The zeros are x = 0 and x = -2

½x(x – 4)½x = 0 and x – 4 = 0x = 0 and x = 4The zeros are x = 0 and x = 4

3x(3x + 1)3x = 0 and 3x + 1 = 0x = 0 and 3x = -1 x = -1/3The zeros are x = 0 and x = -1/3

Holt Algebra 2


A binomial (quadratic expression with two terms) consisting of two perfect squares can be factored using a method called “difference of squares.”

Difference of squares:a2 – b2 = (a + b)(a – b)

Ex) Factor each expression:1) x2 – 9 2) 16x2 – 49

(x + 3)(x – 3) (4x + 7)(4x – 7)

Holt Algebra 2


Find the roots of the equation by factoring.

Example 4A: Find Roots by Using Special Factors

12x2 - 27

3(4x2 – 9)

3(2x – 3)(2x + 3) = 0

2x – 3 = 0 2x + 3 = 0

x = 3/2 x = -3/2

The zeros are x = 3/2 and x = -3/2

Holt Algebra 2


Factor by grouping when you have four terms with no common factor.

Ex) Factor each expression:1)xy – 5y – 2x + 10

2) x2 + 4x – x – 4

y(x – 5) – 2(x – 5)

(y – 2)(x – 5)

x(x + 4) – 1(x + 4)(x – 1)(x + 4)x – 1 = 0 x + 4 = 0x = 1 and x = -4The zeros are x = 1 and x = -4

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No zeros: