1. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 78 § 0.4 Zeros of Functions – The Quadratic Formula and Factoring

2. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 2 of 78  Zeros of Functions  Quadratic Formula  Graphs of Intersecting Lines  Factoring Section Outline

3. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 3 of 78 Zeros of Functions DefinitionExample Zero of a Function: For a function f (x), all values of x such that f (x) = 0.

4. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 4 of 78 Quadratic Formula DefinitionExample Quadratic Formula: A formula for solving any quadratic equation of the form. The solution is: There is no solution if These are the solutions/zeros of the quadratic function

5. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 5 of 78 Graphs of Intersecting FunctionsEXAMPLE SOLUTION Find the points of intersection of the pair of curves. The graphs of the two equations can be seen to intersect in the following graph. We can use this graph to help us to know whether our final answer is correct.

6. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 6 of 78 Graphs of Intersecting Functions To determine the intersection points, set the equations equal to each other, since they both equal the same thing: y. This is the equation to solve. Now we solve the equation for x using the quadratic formula. CONTINUED Subtract x from both sides. Add 9 to both sides. Use the quadratic formula. We now recognize that, for the quadratic formula, a = 1, b = -11, and c =18. Simplify.

7. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 7 of 78 Graphs of Intersecting FunctionsCONTINUED We now find the corresponding y-coordinates for x = 9 and x = 2. We can use either of the original equations. Let’s use y = x – 9. Simplify. Rewrite. Simplify.

8. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 8 of 78 Graphs of Intersecting FunctionsCONTINUED Therefore the solutions are (9, 0) and (2, -7). This seems consistent with the two intersection points on the graph. A zoomed in version of the graph follows.

9. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 9 of 78 FactoringEXAMPLE SOLUTION Factor the following quadratic polynomial. This is the given polynomial. Factor 2x out of each term. Rewrite 3 as Now I can use the factorization pattern: a 2 – b 2 = (a – b)(a + b). Rewrite

10. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 10 of 78 FactoringEXAMPLE SOLUTION Solve the equation for x. This is the given equation. Multiply everything by the LCD: x 2. Distribute. Multiply. Subtract 5x + 6 from both sides. Factor.

11. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 11 of 78 Factoring Set each factor equal to zero. CONTINUED Solve. Since no denominator from the original equation is zero when x = -1 or when x = 6, these are our solutions.