Chapter 7 Quadratic Equations and Functions Algebra 2 Chapter 7 Quadratic Equations and Functions

7-7 Writing Quadratic Equations and Functions WARMUP: Solve the following:

7-7 Writing Quadratic Equations and Functions GOAL: To learn the relationship between the roots and coefficients of a quadratic equation. To write a quadratic equation or function using information about the roots or the graph.

7-7 Writing Quadratic Equations and Functions Our goal, in a nutshell, is to be able to write a quadratic equation if given two roots. Let’s solve this quadratic equation: 2x2 + 2x – 12 = 0 It is factorable: 2(x2 + x – 6) = 0

7-7 Writing Quadratic Equations and Functions 2(x2 + x – 6) = 0 It is still factorable: 2(x + 3)(x – 2) = 0 So: x = -3 or x = 2.

7-7 Writing Quadratic Equations and Functions So let’s work backwards. Say we are given that the roots to some quadratic equation are: x = -3 or x = 2 We can easily reason that our equation is: (x + 3)(x – 2) = 0 And then in standard quadratic form: x2 + x – 6 = 0 There may have been a coefficient factored out, so: a(x2 + x – 6) = 0, a≠0

7-7 Writing Quadratic Equations and Functions By that thinking, we can generalize this: If r1 and r2 are the roots of a quadratic equation, then: (x – r1)(x – r2) = 0 If we foil this, we get: x2 – (r1 + r2)x + r1r2 = 0 And this is a theorem:

7-7 Writing Quadratic Equations and Functions Theorem: A quadratic equation with roots r1 and r2 is: x2 – (r1 + r2)x + r1r2 = 0 or a[x2 – (r1 + r2)x + r1r2]= 0

7-7 Writing Quadratic Equations and Functions Example: Find a quadratic equation with roots 5 and -1: r1 = 5 and r2 = -1 x2 – (r1 + r2)x + r1r2 = 0 so x2 – (5 + -1)x + (5)(-1) = 0 x2 – 4x – 5 = 0

7-7 Writing Quadratic Equations and Functions Another example. Let’s look at example 1 in the book, on page 339:

7-7 Writing Quadratic Equations and Functions Theorem: If r1 and r2 are the roots of a quadratic equation ax2 + bx + c = 0, then r1 + r2 = sum of roots = and r1r2 = product of roots = This theorem is useful when checking solutions.

7-7 Writing Quadratic Equations and Functions More examples.

7-7 Writing Quadratic Equations and Functions HOMEWORK!