 # 2-3 solving quadratic equations by graphing and factoring

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2-3 solving quadratic equations by graphing and factoring
Chapter 2 2-3 solving quadratic equations by graphing and factoring

Objectives Students will be able to:
Solve quadratic equations by graphing or factoring. Determine the quadratic function from its roots.

Zero of a function What is the zero of a function? A zero of a function is a value of the input x that makes the output f(x) equal zero. The zeros of a function are the x-intercepts.

How many solutions or zeros does a quadratic function has? Unlike linear functions, which have no more than one zero, quadratic functions can have two zeros, as shown at next page. These zeros are always symmetric about the axis of symmetry

Example 1 Find the zeros of f(x) = x2 – 6x + 8 by using a graph and table.

Example 1 solution The table and the graph indicate that the zeros are 2 and 4.

Example 2 Find the zeros of g(x) = –x2 – 2x + 3 by using a graph and a table. Enter y = –x2 – 2x + 3 into a graphing calculator.

Example 2 solution Both the table and the graph show that y = 0 at x = –3 and x = 1. These are the zeros of the function.

Student Guided practice
Lets do the quadratic equations worksheet

The roots of an equation
Besides zeros what is another name for the solutions of a quadratic equation? The solution to a quadratic equation of the form ax2 + bx + c = 0 are roots. The roots of an equation are the values of the variable that make the equation true.

Another way we can find the solution of a quadratic equation is called factoring. You can find the roots of some quadratic equations by factoring and applying the Zero Product Property. Remember: Functions have zeros or x-intercepts. Equations have solutions or roots.

Zero product property

Example 3 Find the zeros of the function by factoring.
f(x) = x2 – 4x – 12 Solution: first we set up the equation =0. 𝑥2 – 4𝑥 – 12 = 0. Second we distribute the x’s (𝑥 ) (𝑥 )

Example 3 continue (𝑥 + 2)(𝑥 – 6) = 0 then we solve for x both factors
Third we factor the coefficient and we think of two numbers that multiply we get the last number and add those two numbers we get the middle number. (𝑥 + 2)(𝑥 – 6) = 0 then we solve for x both factors 𝑥 + 2 = 0 𝑜𝑟 𝑥 – 6 = 0 x= –2 or x = 6

Example 4 find the solution to the equation 𝑦=3 𝑥 2 +5𝑥−12

Binomials and trinomials
Quadratic expressions can have one, two or three terms, such as –16t2, – 16t2 + 25t, or –16t2 + 25t Quadratic expressions with two terms are binomials. Quadratic expressions with three terms are trinomials. Some quadratic expressions with perfect squares have special factoring rules.

Special rule

Example 5 Find the roots of the equation by factoring. 18x2 = 48x – 32

Example 6 Find the roots of the equation by factoring. x2 – 4x = –4

Example 6 Solve the equation by factoring. (k + 1)(k − 5) = 0

Student guided practice
Lets do problems from worksheet 2-6

Example 7 Write a quadratic function in standard form with zeros 4 and – 7.

Example 8 Write a quadratic function in standard form with zeros 5 and – 5.

Homework Do problems 2-10, 12,15 from page 82 in the book

closure Today we saw how we can solve quadratic equations by graphing and factoring.