Section 1.5 Notes
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Transcript Section 1.5 Notes
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
11 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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