S OLVING Q UADRATIC E QUATIONS U SING G RAPHS The solutions, or roots, of ax 2 + bx + c = 0 are the x-intercepts. S OLVING Q UADRATIC E QUATIONS G RAPH ICALLY Write the equation in the form ax 2 + bx + c = 0. Write the related function y = ax 2 + bx + c. Sketch the graph of the function y = ax 2 + bx + c. STEP 1 STEP 2 STEP 3 The solution of a quadratic equation in one variable x can be solved or checked graphically with the following steps:

Solve x 2 = 8 algebraically. 1 2 Check your solution graphically. 1 2 x 2 = 8 S OLUTION Write original equation. x 2 = 16 Multiply each side by 2. Find the square root of each side. x =  4 Check these solutions using a graph. Checking a Solution Using a Graph C HECK

Checking a Solution Using a Graph Write the equation in the form ax 2 + bx + c = x 2 = 8 Rewrite original equation. 1 2 x 2 – 8 = 0 Subtract 8 from both sides. y = 1 2 x 2 – 8 Write the related function y = ax 2 + bx + c. 1 2 Check these solutions using a graph. C HECK

Checking a Solution Using a Graph 2 3 Check these solutions using a graph. Sketch graph of y = 2 x 2 – 8. 1 The x-intercepts are  4, which agrees with the algebraic solution. Write the related function y = 1 2 x 2 – 8 y = ax 2 + bx + c. C HECK 4, 0– 4, 0

Solving an Equation Graphically Solve x 2 – x = 2 graphically. Check your solution algebraically. S OLUTION Write the equation in the form ax 2 + bx + c = 0 x 2 – x = 2 Write original equation. x 2 – x – 2 = 0 Subtract 2 from each side. Write the related function y = ax 2 + bx + c. y = x 2 – x – 2 1 2

Solving an Equation Graphically Write the related function y = ax 2 + bx + c. y = x 2 – x – 2 Sketch the graph of the function y = x 2 – x – 2 From the graph, the x-intercepts appear to be x = –1 and x = – 1, 0 2, 0

Solving an Equation Graphically From the graph, the x-intercepts appear to be x = –1 and x = 2. You can check this by substitution. Check x = –1: Check x = 2: x 2 – x = 2 (–1) 2 – (–1) 2 = ? = 2 x 2 – x = 2 4 – 2 = – 2 = 2 ? – 1, 0 2, 0 C HECK

Comparing Two Quadratic Models U SING Q UADRATIC M ODELS IN R EAL L IFE A shot put champion performs an experiment for your math class. Assume that both times he releases the shot with the same initial speed but at different angles. The path of each put can be modeled by one of the equations below, where x represents the horizontal distance (in feet) and y represents the height (in feet). Initial angle of 35º: y = – x x + 6 Initial angle of 65º: y = – x x + 6 Which angle results in a farther throw?

Comparing Two Quadratic Models S OLUTION Begin by graphing both models in the same coordinate plane. Use only positive x -values because x represents the distance of the throw. Which angle results in a farther throw?

Comparing Two Quadratic Models P ARABOLIC M OTION An angle of 45 º produces the maximum distance when an object is propelled from ground level. An angle smaller than 45 º is better when the shot is released above the ground. If the shot is released from 6 feet above the ground, a 43 º angle produces a maximum distance of feet. 65º 45º 43º 35º