Introduction Completing the square can be a long process, and not all quadratic expressions can be factored. Rather than completing the square or factoring, we can use a formula that can be derived from the process of completing the square. This formula, called the quadratic formula, can be used to solve any quadratic equation in standard form, ax 2 + bx + c = : Applying the Quadratic Formula

Key Concepts A quadratic equation in standard form, ax 2 + bx + c = 0, can be solved for x by using the quadratic formula: Solutions of quadratic equations are also called roots. The expression under the radical, b 2 – 4ac, is called the discriminant. The discriminant tells us the number and type of solutions for the equation : Applying the Quadratic Formula

Key Concepts, continued : Applying the Quadratic Formula Discriminant Number and type of solutions NegativeTwo complex solutions 0One real, rational solution Positive and a perfect squareTwo real, rational solutions Positive and not a perfect square Two real, irrational solutions

Common Errors/Misconceptions not setting the quadratic equation equal to 0 before determining the values of a, b, and c forgetting to use ± for problems with two solutions forgetting to change the sign of b dividing by a or by 2 instead of by 2a not correctly following the order of operations not fully simplifying solutions : Applying the Quadratic Formula

Guided Practice Example 2 Use the discriminant of 3x 2 – 5x + 1 = 0 to identify the number and type of solutions : Applying the Quadratic Formula

Guided Practice: Example 2, continued 1.Determine a, b, and c. a = 3, b = –5, and c = : Applying the Quadratic Formula

Guided Practice: Example 2, continued 2.Substitute the values for a, b, and c into the formula for the discriminant, b 2 – 4ac. b 2 – 4ac = (–5) 2 – 4(3)(1) = 25 – 12 = 13 The discriminant of 3x 2 – 5x + 1 = 0 is : Applying the Quadratic Formula

Guided Practice: Example 2, continued 3.Use what you know about the discriminant to determine the number and type of solutions for the quadratic equation. The discriminant, 13, is positive, but it is not a perfect square. Therefore, there will be two real, irrational solutions : Applying the Quadratic Formula ✔

Guided Practice: Example 2, continued : Applying the Quadratic Formula

Guided Practice Example 3 Solve 2x 2 – 5x = 12 using the quadratic formula : Applying the Quadratic Formula

Guided Practice: Example 3, continued 1.Write the quadratic in standard form : Applying the Quadratic Formula 2x 2 – 5x = 12Original equation 2x 2 – 5x – 12 = 0 Subtract 12 from both sides.

Guided Practice: Example 3, continued 2.Determine the values of a, b, and c. a = 2, b = –5, and c = – : Applying the Quadratic Formula

Guided Practice: Example 3, continued 3.Substitute the values of a, b, and c into the quadratic formula : Applying the Quadratic Formula Quadratic formula Substitute values for a, b, and c. Simplify.

Guided Practice: Example 3, continued : Applying the Quadratic Formula

Guided Practice: Example 3, continued 4.Determine the solution(s). Since the discriminant, 121, is positive and a perfect square, there are two real, rational solutions. Write the fraction from step 3 as two fractions and simplify : Applying the Quadratic Formula

Guided Practice: Example 3, continued The solutions to the equation 2x 2 – 5x = 12 are x = 4 or : Applying the Quadratic Formula ✔

Guided Practice: Example 3, continued : Applying the Quadratic Formula