Warm UP Solve the following quadratic Inequality 5-2x2 ≥ -3x

Unit 5: Quadratic Functions Section 11: Nonlinear systems

Essential Question What methods can you use to solve a system that includes a linear equation and a quadratic equation?

Standards in this section Text book pages: P548-555 MCC9-12.A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

Vocabulary Nonlinear system of equations- a system in which at least one of the equations is non linear.

A system made up of a linear equation and a quadratic equation can have no solution, one solution, or two solutions, as shown below.

Example 1: Solving a Nonlinear System by Graphing Solve the system by graphing. Check your answer. y = x2 + 4x + 3 y = x + 3 Step 1 Graph y = x2 + 4x + 3. The axis of symmetry is x = –2. The vertex is (–2, –1). The y-intercept is 3. Another point is (–1, 0).

The substitution method is a good choice when either equation is solved for a variable, both equations are solved for the same variable, or a variable in either equation has a coefficient of 1 or -1. Remember!

Example 2: Solving a Nonlinear system by substitution. Solve the system by substitution. y = x2 - x - 5 y = -3x + 3 Both equations are solved for y, so substitute one expression for y into the other equation for y. -3x + 3 = x2 –x -5 Substitute -3x = 3 for y in the first equation

Check It Out! Example 2 1. Solve the system by substitution. Check your answer. y = 3x2 - 3x + 1 y = -3x + 4 Both equations are solved for y, so substitute one expression for y into the other equation for y. -3x + 4 = 3x2 - 3x + 1 Subtract -3x + 4 for y in first equation. 0 = 3x2 - 3 Subtract -3x + 4 from both sides

Example 3 : Solving a Nonlinear System 3x - y = 1 y = x2 + 4x - 7 A

Check It Out! Example 3 1. Solve each system by elimination. Check your answers.. 2x - y = 2 y = x2 - 5 a Write the system to align the y-terms 2x - y = 2 y = x2 - 5 Add to eliminate y 2x = x2 - 3 -2x -2x Subtract 2x from booth sides

Example 4: Physics Application The increasing enrollment at South Ridge High School can be modeled by the equation E(t) = -t2 + 25t + 600, where t represents the number of years after 2010. The increasing enrollment at Alta Vista High School can be modeled by the equation E(t) = 24t + 570. In what year will the enrollments at the two schools be equal?

When t = 0, the ball and elevator are at the same height because they are both at ground level. Helpful Hint

Examples Solve each system. y = x2 - 4x + 3 y = x - 1 (1, 0), (4, 3) 1. y = 2x2 - 9x - 5 y = -3x + 3 (-1, 6), (4, -9) 2.

Examples y = x2 + 2x - 3 x - y = 5 no solution 3. y = x2 - 7x + 10 2x - y = 8 (3, -2), (6, 4) 4.

Homework Text book: (Exercises 16-4) P 552 # 1-9 Worksheets: Nonlinear systems practice I, II, and III Coach book: p 236-237 #1-9