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 ax 2 + 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 b 2 – 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 ax 2 + bx + c = 0 The discriminant D = b 2 – 4ac Three cases: 1.If D > 0 then there are two real roots or solutions 2.If D = 0 then there is one double real root or solution 3.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: ax 2 + 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 solutions If the solutions are real they will correspond to the x-intercepts when you graph the quadratic function y = ax 2 + bx + c. Ex:
Additional Examples and Resources Additional example of using the discriminant Additional example of using the quadratic formula Additional example of using the quadratic formula Additional example of graphing Web Resources Wikipedia: Quadratic Equation Purple Math: Quadratic Formula Explained Quadratic Equation Calculator
Question 1 If the discriminant for a quadratic equation is 0 then the equation has… 1.2 real solutions2 real solutions 2.1 real solution1 real solution 3.2 complex solutions2 complex solutions 4.no solutionsno solutions Next question
Question 2 Find the discriminant and identify the number and types of solutions for the following: -2r 2 + 5r – 8 = - 2 1.D = -39, two imaginary solutionsD = -39, two imaginary solutions 2.D = 23, two real solutionsD = 23, two real solutions 3.D = -23, two imaginary solutionsD = -23, two imaginary solutions 4.D = -39, two real solutionsD = -39, two real solutions Next question
Question 3 Which equation is correctly written in standard form? 1.2x – 4x 2 – 5 = 02x – 4x 2 – 5 = 0 2.4x 2 + 3x = 64x 2 + 3x = 6 3.5x 2 – 2x + 5 = x5x 2 – 2x + 5 = x 4.X 2 – 2x + 5 = 0X 2 – 2x + 5 = 0 Next question
Question 4 Solve the following quadratic equation for x using the quadratic formula 3x 2 – 5x – 2 = 0 1. 2. 3. 4. Next question
Question 5 Use a calculator to graph the quadratic equation and determine the number of real solutions: 1.One real solutionOne real solution 2.Two real solutionsTwo real solutions 3.Two complex solutionsTwo complex solutions 4.One real and one complex solutionOne real and one complex solution 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
Additional Example: Discriminant First put the equation into standard form Identify a, b, and c Use the formula for the discriminant So this equation has two complex solutions!
Additional Example: 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
Additional Example: Calculator If the solutions are complex then the graph of quadratic function y = ax 2 + bx + c will have no x-intercepts Ex: Does not cross x-axis