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Unit 1
Expressions,
Equations and
Inequalities
1.5 Quadratic Equations
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Objectives:
•
•
•
•
Solve quadratic equations by factoring.
Solve quadratic equations by the square root property.
Solve quadratic equations using the quadratic formula.
Solve equations reducible to quadratic form.
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Definition of a Quadratic Equation
A quadratic equation in x is an equation that can be written
in the general form
ax 2  bx  c  0
where a, b, and c are real numbers, with a  0
A quadratic equation in x is also called a second-degree
polynomial equation in x.
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The Zero-Product Principle
To solve a quadratic equation by factoring, we apply the
zero-product principle which states that:
If the product of two algebraic expressions is zero, then
at least one of the factors is equal to zero.
If AB = 0, then A = 0 or B = 0.
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Solving a Quadratic Equation by Factoring
1. If necessary, rewrite the equation in the general form
ax 2  bx  c  0 , moving all nonzero terms to one side,
thereby obtaining zero on the other side.
2. Factor completely.
3. Apply the zero-product principle, setting each factor
containing a variable equal to zero.
4. Solve the equations in step 3.
5. Check the solutions in the original equation.
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Example: Solving Quadratic Equations by Factoring
Solve by factoring: 2 x 2  x  1
Step 1 Move all nonzero terms to one side and
obtain zero on the other side.
2 x2  x  1  0
Step 2 Factor
(2 x  1)( x  1)  0
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Example: Solving Quadratic Equations by Factoring
(continued)
Steps 3 and 4 Set each factor equal to zero and solve
the resulting equations.
(2 x  1)( x  1)  0
2x 1  0
2x  1
1
x
2
x 1  0
x  1
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Example: Solving Quadratic Equations by Factoring
(continued)
Step 5 Check the solutions in the original equation.
2 x2  x  1
Check
1
x  1
x
2
2
2(1)  1  1?
2
1 1

2     1?
2

1

1

1

1
2 2
1 1
 11 1
2 2
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Solving Quadratic Equations by the Square Root Property
Quadratic equations of the form u2 = d, where u is an
algebraic expression and d is a nonzero real number,
can be solved by the Square Root Property:
If u is an algebraic expression and d is a nonzero real
number, then u2 = d has exactly two solutions:
u d
u d
or
Equivalently,
If u2 = d, then u   d
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Example: Solving Quadratic Equations by the Square
Root Property
Solve by the square root property:
5 x  45  0
2
5 x 2  45
x 2  9
x   9
x  3i
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Example:
Solve the equation:
x  4x 1  0
2
x  2  5
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The Quadratic Formula
The solutions of a quadratic equation in general form
ax 2  bx  c  0 with a  0 , are given by the quadratic
formula:
b  b 2  4ac
x
2a
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Example: Solving a Quadratic Equation Using the
Quadratic Formula
Solve using the quadratic formula:
2x  2x 1  0
a = 2, b = 2, c = – 1
2
b  b 2  4ac
x
2a
(2)  (2) 2  4(2)(1)
x
2(2)
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Example: Solving a Quadratic Equation Using the
Quadratic Formula (continued)
2  4  8
x
4
2  12
x
4
x

2 1  3

4
1  3
x
2
2  2 3
x
4
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