4-4 Factoring Quadratic Expressions

Quick Review

To factor an expression of the form a x 2 + b x + c , eh , x squared , plus b x plus c comma  when a ≠ 1, you find numbers that have the product ac and sum b. You can also factor an expression using the FOIL method in reverse or by finding the greatest common factor (GCF).

Example

Factor the expression 5 x 2 + 13 x + 6 . 5 x squared , plus 13 x plus 6 .

Exercises

Factor each expression.

27. x 2 − 8 x + 12 x squared , minus 8 x plus 12
28. 3 x 2 + 11 x − 20 3 x squared , plus 11 x minus 20
29. − 4 x 2 + 14 x − 6 negative 4 , x squared , plus 14 x minus 6
30. x 2 + 14 x + 40 x squared , plus 14 x plus 40

Factor each perfect square trinomial.

31. x 2 − 14 x + 49 x squared , minus 14 x plus 49
32. 9 x 2 + 30 x + 25 9 x squared , plus 30 x plus 25

Factor each difference of two squares.

33. 36 x 2 − 16 36 x squared , minus 16
34. 25 x 2 − 4 25 x squared , minus 4

Find the GCF of each expression. Then factor each expression.

35. 6 x 2 − 24 x 6 x squared , minus 24 x
36. − 14 x 2 − 49 negative 14 , x squared , minus 49

4-5 Solving Quadratic Equations

Quick Review

The zeros of a quadratic function are the solutions of the related quadratic equation. You can find the zeros from a table or from the x-intercepts of the parabola that is the graph of the function. You can also find them by factoring the standard form of a quadratic equation, a x 2 + b x + c = 0 , eh , x squared , plus b x plus c equals 0 comma  and using the Zero-Product Property.

Example

Solve 2 x 2 + 6 x = 8 2 x squared , plus 6 x equals 8  by factoring.

Exercises

Solve each equation by factoring.

37. x 2 = 4 x + 12 x squared , equals 4 x plus 12
38. 2 x 2 − 3 x − 14 = 0 2 x squared , minus 3 x minus 14 equals 0
39. x 2 + 2 x = 8 x squared , plus 2 x equals 8
40. x 2 + 7 x = 18 x squared , plus 7 x equals 18

Solve each equation by graphing.

41. 5 x 2 + 8 x − 13 = 0 5 x squared , plus 8 x minus 13 equals 0
42. 9 − 4 x = 2 x 2 9 minus 4 x equals , 2 x squared
43. x 2 − x = 1 x squared , minus x equals 1
44. x 2 − 2 x − 4 = 0 x squared , minus 2 x minus 4 equals 0

Solve each equation by using a table.

45. x 2 − 6 x + 8 = 0 x squared , minus 6 x plus 8 equals 0
46. 9 x − 14 = 3 x 2 9 x minus 14 equals , 3 x squared
47. x 2 − 5 x + 2 = 0 x squared , minus 5 x plus 2 equals 0
48. 2 x 2 − 12 x = − 16 2 x squared , minus 12 x equals negative 16

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