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Solving Quadratic Equations using the Quadratic Formula

Learn how to solve quadratic equations using the quadratic formula. Practice solving different types of quadratic equations and check your answers.

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Solving Quadratic Equations using the Quadratic Formula

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  1. Preview Warm Up California Standards Lesson Presentation

  2. Warm Up Evaluate for x = –2, y = 3, and z = –1. 1. x2 4 2. xyz 6 3. x2 – yz 4. y – xz 7 1 5. –x 6. z2 – xy 7 2

  3. California Standards 19.0 Students know the quadratic formula and are familiar with its proof by completing the square. 20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations.

  4. In the previous lesson, you completed the square to solve quadratic equations. If you complete the square of ax2 + bx + c = 0, you can derive the Quadratic Formula.

  5. Remember! To add fractions, you need a common denominator.

  6. Additional Example 1A: Using the Quadratic Formula Solve using the Quadratic Formula. 6x2 + 5x – 4 = 0 6x2 + 5x + (–4) = 0 Identify a, b, and c. Use the Quadratic Formula. Substitute 6 for a, 5 for b, and –4 for c. Simplify.

  7. Additional Example 1A Continued Solve using the Quadratic Formula. 6x2 + 5x – 4 = 0 Simplify. Write as two equations. Solve each equation.

  8. Additional Example 1B: Using the Quadratic Formula Solve using the Quadratic Formula. x2 = x + 20 Write in standard form. Identify a, b, and c. 1x2 + (–1x) + (–20) = 0 Use the Quadratic Formula. Substitute 1 for a, –1 for b, and –20 for c. Simplify.

  9. Additional Example 1B Continued Solve using the Quadratic Formula. x2 = x + 20 Simplify. Write as two equations. Solve each equation. x = 5 or x = –4

  10. Helpful Hint You can graph the related quadratic function to see if your solutions are reasonable.

  11. Check It Out! Example 1a Solve using the Quadratic Formula. Check your answer. –3x2 + 5x + 2 = 0 –3x2 + 5x + 2 = 0 Identify a, b, and c. Use the Quadratic Formula. Substitute –3 for a, 5 for b, and 2 for c. Simplify.

  12. x = – or x = 2 Check It Out! Example 1a Continued Solve using the Quadratic Formula. Check your answer. –3x2 + 5x + 2 = 0 Simplify. Write as two equations. Solve each equation.

  13. x = – or x = 2 Check It Out! Example 1a Continued Solve using the Quadratic Formula. Check your answer. Check –3x2 + 5x + 2 = 0

  14. Check It Out! Example 1b Solve using the Quadratic Formula. Check your answer. 2 – 5x2 = –9x (–5)x2 + 9x + (2) = 0 Write in standard form. Identify a, b, and c. Use the Quadratic Formula. Substitute –5 for a, 9 for b, and 2 for c. Simplify

  15. or x = 2 x = – Check It Out! Example 1b Continued Solve using the Quadratic Formula. Check your answer. 2 – 5x2 = –9x Simplify. Write as two equations. Solve each equation.

  16. –5x2 + 9x + 2 = 0 Check –5x2 + 9x + 2 = 0 –5(2)2 + 9(2) + 2 0 –5 + 9 + 2 0 –20 + 18 + 2 0 0 0 + 2 0  0 0 Check It Out! Example 1b Continued Solve using the Quadratic Formula. Check your answer. 

  17. Because the Quadratic Formula contains a square root, the solutions may be irrational. You can give the exact solution by leaving the square root in your answer, or you can approximate the solutions.

  18. Estimate : x ≈ 1.54 or x ≈ –4.54. Additional Example 2: Using the Quadratic Formula to Estimate Solutions Solve x2 + 3x – 7 = 0 using the Quadratic Formula. Check reasonableness

  19. Estimate : x ≈ 3.87 or x ≈ 0.13. Check It Out! Example 2 Solve 2x2 – 8x + 1 = 0 using the Quadratic Formula. Check reasonableness

  20. There is no one correct way to solve a quadratic equation. Many quadratic equations can be solved using several different methods: graphing, factoring, completing the square, using square roots, and using the Quadratic Formula.

  21. Additional Example 3: Solving Using Different Methods Solve x2 – 9x + 20 = 0. Show your work. Use at least two different methods. Check your answer. Method 1 Solve by graphing. Write the related quadratic function and graph it. y = x2 – 9x + 20 The solutions are the x-intercepts, 4 and 5.

  22. Additional Example 3 Continued Solve x2 – 9x + 20 = 0. Show your work. Use at least two different methods. Check your answer. Method 2 Solve by factoring. x2 – 9x + 20 = 0 (x – 4)(x – 5) = 0 Factor. x – 4 = 0 or x – 5 = 0 Use the Zero Product Property. x = 4 or x = 5 Solve each equation.

  23. x2 – 9x + 20 = 0 Check x2 – 9x + 20 = 0 (5)2 – 9(5) + 20 0 (4)2 – 9(4) + 20 0 25 – 45 + 20 0 16 – 36 + 20 0   0 0 0 0 Additional Example 3 Continued Solve x2 – 9x + 20 = 0. Show your work. Use at least two different methods. Check your answer. Check: 4 and 5

  24. Add to both sides. x2 +7x = –10 Check It Out! Example 3a Solve. Show your work and check your answer. x2 + 7x + 10 = 0 Method 3 Solve by completing the square. x2 + 7x + 10 = 0 x2 + 7x = –10 Factor and simplify. Take the square root of both sides.

  25. or x2 + 7x + 10 = 0 Check x2 + 7x + 10 = 0 (–2)2 + 7(–2) + 10 0 (–5)2 + 7(–5) + 10 0 4 – 14 + 10 0 25 – 35 + 10 0   0 0 0 0 Check It Out! Example 3a Continued Solve. Show your work and check your answer. x2 + 7x + 10 = 0 Solve each equation. x = –2 or x = –5

  26. Check It Out! Example 3b Solve. Show your work and check your answer. –14 + x2 – 5x = 0 Method 4 Solve using the Quadratic Formula. x2 – 5x – 14 = 0 1x2–5x –14 = 0 Identify a, b, and c. Substitute 1 for a, –5 for b, and –14 for c. Simplify.

  27. or Check It Out! Example 3b Continued Solve. Show your work and check your answer. –14 + x2 – 5x = 0 Write as two equations. x = 7or x = –2 Solve each equation.

  28. x2 – 5x – 14 = 0 Check 72 – 5(7) – 14 0 –22 – 5(–2) – 14 0 49 – 35 – 14 0 4 + 10 – 14 0 14 – 14 0 14 – 14 0 0 0 0 0 Check It Out! Example 3b Continued Solve. Show your work and check your answer. –14 + x2 – 5x = 0 x2 – 5x – 14 = 0  

  29. Check It Out! Example 3c Solve. Show your work and check your answer. 2x2 + 4x – 21 = 0 Method 1 Solve by graphing. 2x2 + 4x – 21 = y Write the related quadratic function. Divide each term by 2 and graph. The solutions are the x-intercepts and appear to be ≈ 2.4 and ≈ –4.4.

  30. Check y = 2x2 + 4x – 21 using a calculator. Check It Out! Example 3c Continued Solve. Show your work and check your answer. 2x2 + 4x – 21 = 0 Answers x ≈ –4.39 and ≈ 2.39 appear to be reasonable.

  31. Notice that all of the methods in Example 3 on p. 600 produce the same solution, –1 and –6. The only method you cannot use to solve x2 + 7x + 6 = 0 is using square roots. Sometimes one method is better for solving certain types of equations. The table below gives some advantages and disadvantages of the different methods.

  32. Lesson Quiz 1. Solve x2 + x = 12 by using the Quadratic Formula. 2. Solve –3x2 + 5x = 1 by using the Quadratic Formula. 3.Solve 8x2 – 13x – 6 = 0. Use at least 2 different methods. 3, –4 = 0.23, ≈ 1.43

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