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2. Chapter 11 Quadratic Equations

3. 11.1 Review of Solving Equation by Factoring 11.2 The Square Root Property and Completing the Square 11.3 The Quadratic Formula Putting It All Together 11.4Equations in Quadratic Form 11.5Formulas and Applications 11 Quadratic Equations

4. The Quadratic Formula 11.3 The next method we will discuss for solving quadratic equations is the Quadratic Formula. The quadratic formula is derived by completing the square on the general Quadratic equation.

8. Solve using the quadratic formula. Example 1 Solution Solve a Quadratic Equation Using the Quadratic Formula a = 4 b = -2 c = -7 Quadratic Formula Substitute a = 4, b = -2, and c = -7. Perform the operations.

9. Solve using the quadratic formula. Example 2 Solution a = 1 b = -6 c = 25 Subtract 6y and add 25 to both sides. Identify a, b, and c. Quadratic Formula. Substitute a = 1, b = -6, and c = 25. Perform the operations. 36-100= -64.

10. Solve using the quadratic formula. Example 3 Solution Multiply using FOIL. Subtract 3p and add 5 to both sides. a = 9 b = 6 c = 1 Identify a, b, and c. Quadratic Formula Substitute a = 9, b = 6, c = 1. Perform the operations.

11. Solve using the quadratic formula. Example 4 Solution Notice that the equation is already in the right form. However, working with fractions in the quadratic formula would be difficult. Eliminate the fractions by multiplying the equation by 12, the least common denominator of the fractions. a = 3 b = -4 c = 9 Identify a, b, and c. Quadratic Formula Substitute values for a, b and c. Perform operations. 16-108 = -92. Factor out 2 in the numerator. Simplify.

12. Determine the Number and Type of Solutions to a Quadratic Equation Using the Discriminant

13. Find the value of the discriminant. Then, determine the number and type of solution of each equation. Example 5 Solution The equation is already written in the correct form. Identify a, b, and c. a = 1 b = 3 c = -9 Identify a, b, and c. Since the discriminant is positive but not a perfect square, the equation has two Irrational solutions.

14. Find the value of the discriminant. Then, determine the number and type of solution of each equation. Example 6 Solution Write equation in the correct form. Identify a, b, and c. a = 11 b = -9 c = 6 Identify a, b, and c. Since the discriminant is negative, the equation has two nonreal, complex solutions Of the form a + bi and a – bi.

15. Solve an Applied Problem Using the Quadratic Formula A ball is thrown upward from a height of 20ft. The height h of the ball (in feet) t sec After the ball is released is given by a)How long does it take the ball to reach a height of 8 ft? b)How long does it take the ball to hit the ground? Example 7 Solution a) Find the time it takes for the ball to reach a height of 8 ft. Find t when h = 8. Substitute 8 for h. Write in standard form. Divide by -4. Quadratic Formula Substitute a = 4, b=-4 and c=-3.. Perform Operations

16. Since t represents time, t cannot be - ½. We reject that as a solution.