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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. Quadratic Functions and Equations8
Quadratic Functions and Equations
8.1 Quadratic Equations
8.2 The Quadratic Formula
8.3 Applications Involving Quadratic Equations
8.4 Studying Solutions of Quadratic Equations
8.5 Equations Reducible to Quadratic
8.6 Quadratic Functions and Their Graphs
8.7 More About Graphing Quadratic Functions
8.8 Problem Solving and Quadratic Functions
8.9 Polynomial and Rational Inequalities
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3. Solving Using the Quadratic Formula Approximating Solutions
8.2
The Quadratic Formula
Solving Using the Quadratic Formula
Approximating Solutions
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4. Solving Using the Quadratic Formula
Each time we solve by completing the square, the procedure is the same. In mathematics, when a procedure is repeated many times, a formula can often be developed to speed up our work.
If we begin with a quadratic equation in standard form, ax2 + bx + c = 0, and solve by completing the square we arrive at the quadratic formula.
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5. The Quadratic Formula The solutions of ax2 + bx + c = 0, are given by
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6. Example Solution Solve 3x2 + 5x = 2 using the quadratic formula.
First determine a, b, and c:
3x2 + 5x – 2 = 0;
a = 3, b = 5, and c = –2.
Substituting
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7. The solutions are 1/3 and –2. The check is left to the student.
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8. To Solve a Quadratic Equation
If the equation can easily be written in the form ax2 = p or (x + k)2 = d, use the principle of square roots.
If step (1) does not apply, write the equation in the form ax2 + bx + c = 0.
Try factoring using the principle of zero products.
If factoring seems difficult or impossible, use the quadratic formula. Completing the square can also be used, but is slower.
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9. Recall that a second-degree polynomial in one variable is said to be quadratic. Similarly, a second-degree polynomial function in one variable is said to be a quadratic function.
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10. Example Solution Solve x2 + 7 = 2x using the quadratic formula.
First determine a, b, and c:
x2 – 2x + 7 = 0;
a = 1, b = –2, and c = 7.
Substituting
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11. The check is left to the student.
The solutions are
The check is left to the student.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

12. Approximating Solutions
When the solution of an equation is irrational, a rational-number approximation is often useful. This is often the case in real-world applications similar to those found in section 8.3.
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13. Example
Use a calculator to approximate
Solution
Take the time to familiarize yourself with your calculator:
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