Interpret the Discriminant Warm Up Lesson Presentation Lesson Quiz

Warm-Up Evaluate the expression. 1. 8x2 when x = 4 ANSWER 128 2. –4 + x when x = 9 √ ANSWER –1 A ball is kicked into the air from a height of 4.5 feet with an initial velocity of 30 feet per second. What is the height of the ball after 1 second? 3. ANSWER 18.5 ft

Example 1 Use the discriminant to tell whether the equation has two solutions, one solution, or no solution. Equation ax2 + bx + c = 0 Discriminant b2 – 4ac Number of solutions a. 2x2 + 6x + 5 = 0 62 – 4(2)(5) = –4 No solution b. x2 – 7 = 0 02 – 4(1)(– 7) = 28 Two solutions c. 4x2 – 12x + 9 = 0 (–12)2 –4(4)(9) = 0 One solution

Write the equation in standard form. Example 2 Tell whether the equation 3x2 – 7 = 2x has two solutions, one solution, or no solution. SOLUTION STEP 1 Write the equation in standard form. 3x2 – 7 = 2x Write equation. 3x2 – 2x – 7 = 0 Subtract 2x from each side.

Find the value of the discriminant. Example 2 STEP 2 Find the value of the discriminant. b2 – 4ac = (–2)2 – 4(3)(–7) Substitute 3 for a, – 2 for b, and –7 for c. = 88 Simplify. The discriminant is positive, so the equation has two solutions. ANSWER

Guided Practice Tell whether the equation has two solutions, one solution, or no solution. 1. x2 + 4x + 3 = 0 ANSWER two solutions 2. 2x2 – 5x + 6 = 0 ANSWER no solution 3. –x2 + 2x = 1 ANSWER one solution

Find the number of x-intercepts of the graph of y = x2 + 5x + 8. Example 3 Find the number of x-intercepts of the graph of y = x2 + 5x + 8. SOLUTION Find the number of solutions of the equation 0 = x2 + 5x + 8. b2 – 4ac = (5)2 – 4(1)(8) Substitute 1 for a, 5 for b, and 8 for c. –7 = Simplify. The discriminant is negative, so the equation has no solution. This means that the graph of y = x2 + 5x + 8 has no x-intercepts. ANSWER

Example 3 CHECK You can use a graphing calculator to check the answer. Notice that the graph of y = x2 + 5x + 8 has no x-intercepts.

Guided Practice Find the number of x – intercepts of the graph of the function. 4. y = x2 + 10x + 25 ANSWER 1 5. y = x2 – 9x ANSWER 2 6. y =  x2 + 2x  4 ANSWER

Example 4 FOUNTAINS The Centennial Fountain in Chicago shoots a water arc that can be modeled by the graph of the equation y =  0.006x2 + 1.2x + 10 where x is the horizontal distance (in feet) from the river’s north shore and y is the height (in feet) above the river. Does the water arc reach a height of 50 feet? If so, about how far from the north shore is the water arc 50 feet above the water?

Example 4 SOLUTION STEP 1 Write a quadratic equation. You want to know whether the water arc reaches a height of 50 feet, so let y = 50. Then write the quadratic equation in standard form. y 0.006x2 1.2x 10 = – + Write given equation. 50 0.006x2 1.2x 10 = – + Substitute 50 for y. 0 0.006x2 1.2x 40 = – + Subtract 50 from each side.

Find the value of the discriminant of 0 = – 0.006x2 + 1.2x – 40. Example 4 STEP 2 Find the value of the discriminant of 0 = – 0.006x2 + 1.2x – 40. b2 4ac = (1.2)2 4(–0.006)(–40) – = 0.48 a = –0.006, b = 1.2, c = –40 Simplify. STEP 3 Interpret the discriminant. Because the discriminant is positive, the equation has two solutions. So, the water arc reaches a height of 50 feet at two points on the water arc.

Example 4 STEP 4 Solve the equation to find the distance from the north shore to where the water arc is 50 feet above the water. 0 = – 0.006x2 + 1.2x – 40 – x =  b 2 4ac ± b 2a Quadratic formula = –  ± 1.2 0.48 2(–0.006) Substitute values in the quadratic formula. x 42 or x 158 Use a calculator. The water arc is 50 feet above the water about 42 feet from the north shore and about 158 feet from the north shore. ANSWER

Guided Practice 7. WHAT IF? In Example 4, does the water arc reach a height of 70 feet? If so, about how far from the north shore is the water arc 70 feet above the water? Yes; 100 ft ANSWER

Lesson Quiz Tell whether the equation has two solutions, one solution, or no solutions. 1. 4b2 + 2b – 5 = 0 ANSWER two solutions 2. 2g2 + 8g = – 11 ANSWER no solutions Find the number of x-intercepts of the graphs of the equation. 3. y = x2 + 14x + 49 ANSWER one y = x2 + 14x + 50 4. ANSWER none

Lesson Quiz 5. The graph of y = – 0.2x2 + 3.5x models the height of one of the arches at the entrance to a parking structure. Can a truck that is 20 feet high fit under the arch? The value of the discriminate of 0 = – 0.2x2 + 3.5x – 20 is negative, so there is no solution.The truck will not fit under the arch. ANSWER