The Quadratic Formula

Solving quadratic equations

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

Seems hard to remember?Here’s a song to help!

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

Credits

Credits

Kelley, M. (Songwriter & Performer) & Lin, F. (Videoproducer). (2006). The Quadratic Formula [music video]. Retrieved July 10, 2009 from http://www.youtube.com/watch?v=m0h5r5h-Ems