Post on 26-Dec-2015
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Objectives
• You will use the discriminant to find the number and nature of the roots of a quadratic function
• You will solve a quadratic equation by using the quadratic formula
• You will use a graphing calculator to confirm solutions
Why use the quadratic formula?
As you know a quadratic equation is one of the form ax2 + bx + c = 0.
Finding solutions by factoring may be difficult or impossible.
When an equation is in this form we can use the quadratic formula to solve for x.
There may be one real solution, two real solutions, or two complex solutions.
What is the quadratic formula?
To use the quadratic formula make sure the equation is in standard form as below
The Discriminant and Solutions
Notice the part under the radical the b2 – 4ac.This is called the discriminant, because it will
discriminate between the types of solutions we have.
Discriminant: 3 cases
For a quadratic equation in standard form ax2 + bx + c = 0The discriminant D = b2 – 4ac
Three cases:1. If D > 0 then there are two real roots or solutions2. If D = 0 then there is one double real root or solution3. If D < 0 then there are two complex roots or solutions
Example of using discriminant
Example: here a = 2, b = -3, and c = -6.
Plug these values in for the discriminant we get D = (-3)2 – 4(2)(-6) = 57
Since D is bigger than zero that means this equation has two real solutions!
Steps to using the quadratic formula
The discriminant is useful if we only want to know the number and type of solutions. If we want an exact solution, use the formula.
• Make sure the equation is in standard form:• ax2 + bx + c = 0• Identify the coefficients a, b, and c and
substitute into the formula.• Simplify the solution completely including the
radical and any fractions.
Example using quadratic formula• Given equation• Put into standard form• Identify the coefficients• Recall the formula
• Substitute in a, b, c
• Simplify under radical• Simplify the radical and
fractions to get your final answer
Using the calculator to verify solutionsIf the solutions are real they will correspond to the x-intercepts when you graph the quadratic function y = ax2 + bx + c.Ex:
Additional Examples and Resources
• Additional example of using the discriminant• Additional example of using the quadratic for
mula• Additional example of graphingWeb ResourcesWikipedia: Quadratic EquationPurple Math: Quadratic Formula ExplainedQuadratic Equation Calculator
Question 1If the discriminant for a quadratic equation is 0
then the equation has…1.2 real solutions2.1 real solution3.2 complex solutions4.no solutions
Next question
Question 2Find the discriminant and identify the number
and types of solutions for the following:-2r2 + 5r – 8 = - 21.D = -39, two imaginary solutions2.D = 23, two real solutions3.D = -23, two imaginary solutions4.D = -39, two real solutions
Next question
Question 3Which equation is correctly written in standard
form?1.2x – 4x2 – 5 = 02.4x2 + 3x = 63.5x2 – 2x + 5 = x4.X2 – 2x + 5 = 0
Next question
Question 4Solve the following quadratic equation for x
using the quadratic formula3x2 – 5x – 2 = 01. 2. 3. 4.
Next question
Question 5Use a calculator to graph the quadratic equation
and determine the number of real solutions:
1.One real solution2.Two real solutions3.Two complex solutions4.One real and one complex solution
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