Solving Quadratic Equations by Graphing (9-2) Objective: Solve quadratic equations by graphing. Estimate solutions of quadratic equations by graphing.

Solve by Graphing A quadratic equation can be written in the standard form ax 2 + bx + c = 0, where a ≠ 0. To write a quadratic function as an equation, replace y or f(x) with 0. The solutions or roots of an equation can be identified by finding the x-intercepts of the related graph. Quadratic equations may have two, one, or no solutions.

Solve by Graphing Two unique real solutions. One unique real solution. No real solutions.

Example 1 Solve x 2 – 3x – 10 = 0 by graphing. XY x = {-2, 5}

Check Your Progress Choose the best answer for the following. –Solve x 2 – 2x – 8 = 0 by graphing. A.{-2, 4} B.{2, -4} C.{2, 4} D.{-2, -4} y = x 2 – 2x - 8

Solve by Graphing The solutions in Example 1 were two distinct numbers. Sometimes the two roots are the same number, called a double root.

Example 2 Solve x 2 + 8x = -16 by graphing.  x 2 + 8x + 16 = 0 XY x = {-4}

Check Your Progress Choose the best answer for the following. –Solve x 2 + 2x = -1 by graphing. A.{1} B.{-1} C.{-1, 1} D.  x 2 + 2x + 1 = 0 y = x 2 + 2x + 1

Solve by Graphing Sometimes the roots are not real numbers.

Example 3 Solve x 2 + 2x + 3 = 0 by graphing. XY x = 

Check Your Progress Choose the best answer for the following. –Solve x 2 + 4x + 5 = 0 by graphing. A.{1, 5} B.{-1, 5} C.{5} D.  y = x 2 + 4x + 5

Estimate Solutions The real roots found thus far have been integers. However, the roots of quadratic equations are usually not integers. In these cases, use your calculator to approximate the roots, or zeros of the equation.

Example 4 Solve x 2 – 4x + 2 = 0 by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth. XY x = {0.6, 3.4} y = x 2 – 4x + 2

Check Your Progress Choose the best answer for the following. –Solve x 2 – 5x + 1 = 0 by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth. A.0.4, 5.6 B.0.1, 4.9 C.0.2, 4.8 D.0.3, 4.7 y = x 2 – 5x + 1

Solve by Graphing Approximating the x-intercepts of graphs is helpful for real-world applications.

Example 5 Consuela built a model rocket for her science project. The equation h = -16t t models the flight of the rocket launched from ground level at a velocity of 250 feet per second, where h is the height of the rocket in feet after t seconds. Approximately how long was Consuela’s rocket in the air?  y = -16x x  x = seconds

Check Your Progress Choose the best answer for the following. –Martin hits a golf ball with an upward velocity of 120 feet per second. The function h = -16t t models the flight of the golf ball hit at ground level, where h is the height of the ball in feet after t seconds. How long was the golf ball in the air? A.Approximately 3.5 seconds B.Approximately 7.5 seconds C.Approximately 4.0 seconds D.Approximately 6.7 seconds y = -16x x