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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
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2
3
4
5
6
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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
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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
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