OBJECTIVES

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Quadratic Equations

Solve a quadratic equation by factoring.Solve a quadratic equation by the square root method.Solve a quadratic equation by completing the square.Solve a quadratic equation by using the quadratic formula.Solve a quadratic equations with complex solutions.Solve applied problems.

SECTION 1.4

1

2

3

4

5

6

QUADRATIC EQUATION

A quadratic equation in the variable x is an equation equivalent to the equation

where a, b, and c are real numbers and a ≠ 0.

ax2 bx c 0,

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THE ZERO-PRODUCT PROPERTY

Let A and B be two algebraic expressions.

Then AB = 0 if and only if A = 0 or B = 0.

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EXAMPLE 1 Page 128 # 22

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EXAMPLE 2 Page 128 # 26

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Suppose u is any algebraic expression and d ≥ 0.

THE SQUARE ROOT PROPERTY

If u2 d, then u d .

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EXAMPLE 3 Page 128 # 38, # 44 and # 46

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A quadratic trinomial x in with coefficient of x2 equal to 1 is a perfect-square trinomial if the constant term is the square of one-half the coefficient of x.

PERFECT SQUARE TRNOMIAL

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EXAMPLE 4Solving a Quadratic Equation by Completing the Square

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04914. 2 xxa

02110. 2 xxb

Step 1 Rearrange the quadratic equation so that the terms in x2 and x are on the left side of the equation and the constant term is on the right side.

Step 2 Make the coefficient of x2 equal to 1 by dividing both sides of the equation by the original coefficient. (Steps 1and 2 are interchangeable.)

METHOD OF COMPLETING THE SQUARE

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Step 3 Add the square of one-half the coefficient of x to both sides of the equation.

Step 4 Write the equation in the form (x + k)2 = d using the fact that the left side is a perfect square.

METHOD OF COMPLETING THE SQUARE

Step 5 Take the square root of each side, prefixing ± to the right side.

Step 6 Solve the two equations from Step 5.

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EXAMPLE 5 Page 129 # 66

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The solutions of the quadratic equation in the standard form ax2 + bx + c = 0 with a ≠ 0 are given by the formula

THE QUADRATIC FORMULA

2 4.

2

b b acx

a

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EXAMPLE 6 Page 129 # 76 and # 86

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In the quadratic formula

THE DISCRIMINANT

the quantity b2 – 4ac under the radical sign is called the discriminant of the equation.

2

,2

4b cbx

a

a

The discriminant reveals the type of solutions of the equation.

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THE DISCRIMINANT

Discriminant Solutions

b2 – 4ac > 0 Two unequal real

b2 – 4ac = 0 One real

b2 – 4ac < 0 Two nonreal complex

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EXAMPLE 7 Using the Discriminant

Use the discriminant to determine the number and type of solutions of each quadratic equation.

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94#.

90#.

88#.

129

c

b

a

Page