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Transcript
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 1 *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 2 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 2 Example 4) Solve by completing square: x 2 4 x 1 0 . 2 *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 . 2a 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. 3 *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) 4