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Section 1.5 Quadratic Equations
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 . It is also called a second-degree
polynomial equation in x.
*Solving Quadratic Equation by Factoring
The Zero-Product Principle
If AB  0 , then A  0 or B  0 .
where A and B are algebraic expressions.
Example 1) Solve the equation ( x  2)( x  3)  0
Example 2) Solve the following quadratic equations by factoring.
a. 3x 2  9 x  0
Step 1 Move all terms to one side and obtain zero on the other side.
Step 2 Factor.
Step 3 Set each factor equal to zero and solve the resulting equations.
Step 4 Check the solutions in the original equation.
b. 2 x 2  x  1
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*Solving Quadratic Equation by the Square Root Property
The Square Root Property
Suppose that u is an algebraic expression and d is a nonzero real number.
If u 2  d , then u  d or u   d .
Equivalently,
If u 2  d , then u   d .
Example 3) Solve by the square root property
a. 3 x 2  21  0
b. 5 x 2  45  0
c. ( x  5) 2  11
*Completing the Square
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b
If x 2  bx is a binomial, then by adding   , which is the square of the half the
2
coefficient of x, a perfect square trinomial will result. That is,
2
b
b

x 2  bx      x  
2
2

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Example 4) Solve by completing square: x 2  4 x  1  0 .
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*Solving Quadratic Equations Using 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
x
 b  b 2  4ac
.
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Example 5) Solve using the quadratic formula:
a. 2 x 2  2 x  1  0
b. x 2  2 x  2  0
*Discriminant
The quantity b 2  4ac , which appears under the radical sign in the quadratic formula, is
called the discriminant.
Table 1.3 (page 140 on the text book)
Discriminant
Kinds of Solutions to
Graph of
2
2
b  4ac
ax  bx  c  0
y  ax 2  bx  c
Two unequal real solutions
b 2  4ac  0
b 2  4ac  0
One solution (a repeated
solution) that is a real number
b 2  4ac  0
No real solution; two imaginary
solutions; The solutions are
complex conjugates.
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*Determining Which Method to Use
Look at Table 1.4 (page 141 on the text book) and PRACTICE lots of problems.
*Applications
Example 6) The formula P  0.01A 2  0.05 A  107 models a woman’s normal systolic
blood pressure, P, at age A. Use this formula to find the age, to the nearest year, of a
woman whose normal systolic blood pressure is 115 mm Hg.
Example 7) What is the width of a 15-inch television set whose height is 9 inches? (Hint:
Recall the Pythagorean Theorem)
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