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5.3 Zeros of the Quadratic 461 Section 5.3 Zeros of the Quadratic 461 5.3 Zeros of the Quadratic We’ve seen how vertex form and intelligent use of the axis of symmetry can help to draw an accurate graph of the quadratic function defined by the equation f(x) = ax2 +bx+c. When drawing the graph of the parabola it is helpful to know where the graph of the parabola crosses the x-axis. That is the primary goal of this section, to find the zero crossings or x-intercepts of the parabola. Before we begin, you’ll need to review the techniques that will enable you to factor the quadratic expression ax2 + bx + c. Factoring ax2 + bx + c when a = 1 Our intent in this section is to provide a quick review of techniques used to factor quadratic trinomials. We begin by showing how to factor trinomials having the form ax2 + bx + c, where the leading coefficient is a = 1; that is, trinomials having the form x2 +bx+c. In the next section, we will address the technique used to factor ax2 +bx+c when a 6= 1. Let’s begin with an example. 2 I Example 1. Factor x + 16x − 36. Note that the leading coefficient, the coefficient of x2, is a 1. This is an impor- tant observation, because the technique presented here will not work when the leading coefficient does not equal 1. Note the constant term of the trinomial x2 + 16x − 36 is −36. List all integer pairs whose product equals −36. 1, −36 −1, 36 2, −18 −2, 18 3, −12 −3, 12 4, −9 −4, 9 6, −6 −6, 6 Note that we’ve framed the pair −2, 18. We’ve done this because the sum of this pair of integers equals the coefficient of x in the trinomial expression x2 + 16x − 36. Use this framed pair to factor the trinomial. x2 + 16x − 36 = (x − 2)(x + 18) It is important that you check your result. Use the distributive property to multiply. (x − 2)(x + 18) = x(x + 18) − 2(x + 18) = x2 + 18x − 2x − 36 = x2 + 16x − 36 1 Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ Version: Fall 2007 462 Chapter 5 Quadratic Functions Thus, our factorization is correct. Let’s summarize the technique. Algorithm. To factor the quadratic x2 + bx + c, proceed as follows: 1. List all the integer pairs whose product equals c. 2. Circle or frame the pair whose sum equals the coefficient of x, namely b. Use this pair to factor the trinomial. Let’s look at another example. 2 I Example 2. Factor the trinomial x − 25x − 84. List all the integer pairs whose product is −84. 1, −84 −1, 84 2, −42 −2, 42 3, −28 −3, 28 4, −21 −4, 21 6, −14 −6, 14 7, −12 −7, 12 We’ve framed the pair whose sum equals the coefficient of x, namely −25. Use this pair to factor the trinomial. x2 − 25x − 84 = (x + 3)(x − 28) Check. (x + 3)(x − 28) = x(x − 28) + 3(x − 28) = x2 − 28x + 3x − 84 = x2 − 25x − 84 With experience, there are a number of ideas that will quicken the process. • As you are listing the integer pairs, should you happen to note that the current pair has the appropriate sum, there is no need to list the remaining integer pairs. Simply halt the process of listing the integer pairs and use the current pair to factor the trinomial. • Some students are perfectly happy being asked “Can you think of an integer pair whose product is c and whose sum is b (where b and c refer to the coefficients of x2 + bx + c)?” If you can pick the pair “out of the air” like this, all is well and good. Version: Fall 2007 Section 5.3 Zeros of the Quadratic 463 Use the integer pair to factor the trinomial and don’t bother listing any integer pairs. Now, let’s investigate how to proceed when the leading coefficient is not 1. Factoring ax2 + bx + c when a 6= 1 When a 6= 1, we use a technique called the ac-test to factor the trinomial ax2 + bx + c. The process is best explained with an example. 2 I Example 3. Factor 2x + 13x − 24. Note that the leading coefficient does not equal 1. Indeed, the coefficient of x2 in this example is a 2. Therefore, the technique of the previous examples will not work. Thus, we turn to a similar technique called the ac-test. First, compare 2x2 + 13x − 24 and ax2 + bx + c and note that a = 2, b = 13, and c = −24. Compute the product of a and c. This is how the technique earns its name “ac-test.” ac = (2)(−24) = −48 List all integer pairs whose product is ac = −48. 1, −48 −1, 48 2, −24 −2, 24 3, −16 −3, 16 4, −12 −4, 12 6, −8 −6, 8 We’ve framed the pair whose sum is b = 13. The next step is to rewrite the trinomial 2x2 + 13x − 24, splitting the middle term into a sum, using our framed integer pair. 2x2 + 13x − 24 = 2x2 − 3x + 16x − 24 We factor an x out of the first two terms, then an 8 out of the last two terms. This process is called factoring by grouping. 2x2 − 3x + 16x − 24 = x(2x − 3) + 8(2x − 3) We now factor out a common factor of 2x − 3. x(2x − 3) + 8(2x − 3) = (x + 8)(2x − 3) It’s helpful to see the complete process as a coherent unit. Version: Fall 2007 464 Chapter 5 Quadratic Functions 2x2 + 13x − 24 = 2x2 − 3x + 16x − 24 = x(2x − 3) + 8(2x − 3) = (x + 8)(2x − 3) Check. Again, it is important to check the answer by multiplication. (x + 8)(2x − 3) = x(2x − 3) + 8(2x − 3) = 2x2 − 3x + 16x − 24 = 2x2 + 13x − 24 Because this is the original trinomial, our solution checks.2 Let’s summarize this process. Algorithm: ac-Test. To factor the quadratic ax2 + bx + c, proceed as follows: 1. List all integer pairs whose product equals ac. 2. Circle or frame the pair whose sum equals the coefficient of x, namely b. 3. Use the circled pair to express the middle term bx as a sum. 4. Factor by “grouping.” Let’s look at another example. 2 I Example 4. Factor 3x + 34x − 24. Compare 3x2 + 34x − 24 and ax2 + bx + c and note that a = 3, b = 34 and c = −24. List all integer pairs whose product equals ac = (3)(−24) = −72. 1, −72 −1, 72 2, −36 −2, 36 3, −24 −3, 24 4, −18 −4, 18 6, −12 −6, 12 8, −9 −8, 9 We’ve framed the pair whose sum is the same as b = 34, the coefficient of x in 3x2 + 34x − 24. Again, possible shortcuts are possible. If you can “think” of a pair whose product is ac = −72 and whose sum is b = 34, then it is not necessary to list any integer pairs. Alternatively, if you come across the needed pair as you are listing 2 If you check a number of your results, it will soon become apparent why the ac-test works so well. Version: Fall 2007 Section 5.3 Zeros of the Quadratic 465 them, then you can halt the process. There is no need to list the remaining pairs if you have the one you need. Use the framed pair to express the middle term as a sum, then factor by grouping. 3x2 + 34x − 24 = 3x2 − 2x + 36x − 24 = x(3x − 2) + 12(3x − 2) = (x + 12)(3x − 2) We leave it to the reader to check this result. Intercepts The points where the graph of a function crosses the x-axis are called the x-intercepts of graph of the function. Consider the graph of the quadratic function f in Figure 1. y 10 f x (−3; 0) (2; 0) 10 (0; −6) Figure 1. The x- and y-intercepts are key features of any graph. Note that the graph of the f crosses the x-axis at (−3, 0) and (2, 0). These are the x-intercepts of the parabola. Note that the y-coordinate of each x-intercept is zero. In function notation, the solutions of f(x) = 0 (note the similarity to y = 0) are the x-coordinates of the points where the graph of f crosses the x-axis. Analyzing the graph of f in Figure 1, we see that both −3 and 2 are solutions of f(x) = 0. Thus, the process for finding the x-intercepts is clear. Finding x-intercepts. To find the x-intercepts of the graph of any function, set y = 0 and solve for x. Alternatively, if function notation is used, set f(x) = 0 and solve for x. Let’s look at an example. Version: Fall 2007 466 Chapter 5 Quadratic Functions I Example 5. Find the x-intercepts of the graph of the quadratic function defined by y = x2 + 2x − 48. To find the x-intercepts, first set y = 0. 0 = x2 + 2x − 48 Next, factor the trinomial on the right. Note that the coefficient of x2 is 1. We need only think of two integers whose product equals the constant term −48 and whose sum equals the coefficient of x, namely 2. The numbers 8 and −6 come to mind, so the trinomial factors as follows (readers should check this result). 0 = (x + 8)(x − 6) To complete the solution, we need to use an important property of the real numbers called the zero product property.
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