# 11-2 Solving Quadratic Equations by Graphing SOLVING QUADRATIC EQUATIONS BY GRAPHING You will learn 5 different ways to solve a quadratic equations. 1. Solve by graphing 2. Solve by factoring 3. Solve using square roots 4. Solve by completing the square 5. Solve using the quadratic equation Quadratic equations can have: Two distinct solutions (roots, zeros) One distinct solutions (roots, zeros) No real solutions (roots, zeros) They can be found by graphing the equation to see where

the parabola crosses the x-axis (x intercepts). y y y

x x

x

Two solutions One Solution No Solutions

Find the solutions of the quadratic function from its graph. y = x2 2x 3 y = x2 + 8x + 16 2) 1) x = -1 and x=3

x = -4 Find the solutions of the quadratic function from its graph. y = x2 6x + 9 y = 2x 2 2 3) 4) NO SOLUTION

x=3 Before graphing You must put all the terms on one side of the equals sign. The other side should = 0! **Always use inverse operations to move pieces in an equation. REVIEW: Graphing a quadratic function. Steps: 1. Find the axis of symmetry & graph (dashed lines or highlighter) 2. Find the vertex & graph it 3. Find the y-intercept & graph it 4. Find two points on the left of the axis of symmetry 5. Reflect them over the axis of symmetry

You can do this because quadratics are symmetrical 6. Connect points with a smooth curve and arrows on the end. This will be a total of 6 points! Find the solutions of each quadratic equation. 5) A. of S. 0 8 1 3 Vertex: 2 0 3 -1 Y-intercept: 4 0 Solutions:

x = 2, x = 4 6x + 8 = x2 Find the solutions of each quadratic equation. 6) A. of S. -2 0 Vertex: -1 -6 0 -8 Y-intercept: 1 -6 2 0 Solutions: x = -2, x = 2

2x2 8 = 0 Find the solutions of each quadratic equation. 7) A. of S. -3 6 Vertex: -2 0 -1 -2 Y-intercept: 0 0 1 6 Solutions: x = -2, x = 0

2x2 + 4x 3 = 3 Find the solutions of each quadratic equation. 8) A. of S. -5 -8 Vertex: -4 -4 -3 -2 Y-intercept: -2 -2 -1 -4 Solutions: NO SOL. x2 5x 8 = 2x2

Finding on your calculator This is really helpful to know how to do when you get to a problem that does not have a whole number solution!! Use your calculator to find the solution(s) for each quadratic equation. 9) 11) x = 2 and x = -2

10) x = 2 and x = -4 NO SOL. 12) 66 x -4.58 and x 4.58 13) A dolphin jumps out of the water. The quadratic function y = 16x2 + 32 x models the dolphins height above the water after x seconds. About how long is the dolphin out of the water?

y = 16x2 + 32 x Dolphin was in the air for 2 seconds! 14) A frog jumps straight up from the ground. The quadratic function f(t) = 16t2 + 12t models the frogs height above the ground after t seconds. About how long is the frog in the air? f(t) = 16t2 + 12t

The frog was in the air for less than a second. The frog was in the air for: 0.75 seconds