The Discriminant Lesson 9.9

Warm Up 1. x2 – 5x – 6 = 0 2. 2x2 + 2x – 24 = 0 3. x2 + 10x + 25 = 0 Use the Quadratic Formula to solve each equation. 1. x2 – 5x – 6 = 0 2. 2x2 + 2x – 24 = 0 3. x2 + 10x + 25 = 0 1, –6 3, –4 0, –5

California Standards 22.0 Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points. 23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.

Objective Students determine the number of solutions of a quadratic equation by using the discriminant.

If the quadratic equation is in standard form, its discriminant is b2 – 4ac. Notice that this is the expression under the square root in the Quadratic Formula. Recall that quadratic equations can have two, one, or no real solutions. You can determine the number of solutions of a quadratic equation by evaluating the discriminant.

positive negative

Find the number of solutions of 3x2 – 2x + 2 = 0 by using the discriminant. a = 3, b = –2, c = 2 Identify the values of a, b, and c. b2 – 4ac = (–2)2 – 4(3)(2) Substitute 3, –2, and 2 for a, b, and c. = 4 – 24 Simplify. = –20 b2 – 4ac is negative. There are no real solutions.

Find the number of solutions of 2x2 + 11x + 12 = 0 by using the discriminant. a = 2, b = 11, c = 12 Identify the values of a, b, and c. Substitute 2, 11, and 12 for a, b, and c. b2 – 4ac = 112 – 4(2)(12) = 121 – 96 Simplify. = 25 b2 – 4ac is positive. There are two solutions.

Recall that the solutions to a quadratic are the same as the x-intercepts of the related function. The discriminant can be used to find the number of x-intercepts.

Find the number of x-intercepts of y = 5x2 + 3x + 1 by using the discriminant. a = 5, b = 3, c = 1 Identify the values of a, b, and c. Substitute 5, 3, and 1 for a, b, and c. b2 – 4ac = 32 – 4(5)(1) = 9 – 20 Simplify. = –11 b2 – 4ac is negative. Therefore, the function y = 5x2 + 3x + 1 has no x-intercepts. The graph does not intercept the x-axis.

Find the number of x-intercepts of y = x2 – 9x + 4 by using the discriminant. a = 1, b = –9, c = 4 Identify the values of a, b, and c. Substitute 1, –9, and 4 for a, b, and c. b2 – 4ac = (–9)2 – 4(1)(4) = 81 – 16 Simplify. = 65 b2 – 4ac is positive. Therefore, the function y = x2 – 9x + 4 has two x-intercepts. The graph intercepts the x-axis twice.

Physical Science Application The height h in feet of an object shot straight up with initial velocity v in feet per second is given by h = –16t2 + vt + c, where c is the initial height of the object above the ground. The ringer on a carnival strength test is 2 feet off the ground and is shot upward with an initial velocity of 30 feet per second. Will it reach a height of 20 feet? Use the discriminant to explain your answer.

…Continued h = –16t2 + vt + c 20 = –16t2 + 30t + 2 Substitute 20 for h, 30 for v, and 2 for c. 20 = –16t2 + 30t + 2 0 = –16t2 + 30t + (–18) Subtract 20 from both sides. b2 – 4ac Evaluate the discriminant. 302 – 4(–16)(–18) = –252 Substitute –16 for a, 30 for b, and –18 for c. The discriminant is negative, so there are no real solutions. The ringer will not reach a height of 20 feet.

Lesson Quiz 1. Find the number of solutions of 5x2 – 19x – 8 = 0 by using the discriminant. 2. Find the number of x-intercepts of y = –3x2 + 2x – 4 by using the discriminant. 3. An object is shot up from 4 ft off the ground with an initial velocity of 48 ft/s. Will it reach a height of 40 ft? Use the discriminant to explain your answer. 2 2 The discriminant is 0. The object will reach its maximum height of 40 ft once.