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## Topic 7.3.1

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1. The Quadratic Formula Topic 7.3.1

2. Topic 7.3.1 The Quadratic Formula 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. What it means for you: You’ll use the quadratic formula to solve quadratic equations — and you’ll derive the quadratic formula itself. • Key words: • quadratic formula • completing the square

3. Topic 7.3.1 The Quadratic Formula You can also use the quadratic formula to solve quadratic equations. It works every time.

4. ax2 + bx + c = 0 (a¹ 0) Topic 7.3.1 The Quadratic Formula Quadratic Equations can be in Any Variable The standard form for a quadratic equation is: Any quadratic equation can be written in this form by, if necessary, rearranging it so that zero is on one side. A lot of the quadratic equations you will see may contain a variable other than x — but they are still quadratic equations like the one above, and can be solved in the same way.

5. Topic 7.3.1 The Quadratic Formula Example 1 Find the solutions of the following quadratic equations:a) x2 – 4x + 3 = 0 b) y2 – 4y + 3 = 0 Solution a) Here, a = 1, b = –4, and c = 3. This equation factors to give (x – 3)(x – 1) = 0 So using the zero property:(x – 3) = 0 or (x – 1) = 0, or x = 3 or x = 1 Solution continues… Solution follows…

6. Topic 7.3.1 The Quadratic Formula Example 1 Find the solutions of the following quadratic equations:a) x2 – 4x + 3 = 0 b) y2 – 4y + 3 = 0 Solution (continued) b) Here, the variable is y rather than x— but that does not affect the solutions. Againa = 1, b = –4, and c = 3, so it is the same equation as in a), and will have the same solutions: y = 3or y = 1 You can see that the two quadratic equations are really the same — only the variables have changed.

7. Topic 7.3.1 The Quadratic Formula The Quadratic Formula Solves Any Quadratic Equation The solutions to the quadratic equation ax2 + bx + c = 0 are given by the quadratic formula: You can derive the formula by completing the square.

8. ax2 + x = – Topic 7.3.1 The Quadratic Formula 1. Start with the standard form 2. Subtract c ax2 + bx + c = 0 ax2 + bx = –c 3. Divide the equation by a 4. Complete the square 5. Add the fractions 6. Simplify 7. Include positive and negative roots 8. Rearrange to give the quadratic formula

9. Topic 7.3.1 The Quadratic Formula Example 2 Solve x2 – 5x – 14 = 0 using the quadratic formula. Solution Start by writing down the values of a, b, and c: a = 1, b = –5, and c = –14 Now very carefully substitute these into the quadratic formula. Then simplify to find the values of x. Solution continues… Solution follows…

10. So or Topic 7.3.1 The Quadratic Formula Example 2 Solve x2 – 5x – 14 = 0 using the quadratic formula. Solution (continued)

11. Topic 7.3.1 The Quadratic Formula Example 3 Solve 2x2 – 3x – 2 = 0 using the quadratic formula. Solution Start by writing down the values of a, b, and c: a = 2, b = –3, and c = –2 Now very carefully substitute these into the quadratic formula. Then simplify to find the values of x. Solution continues… Solution follows…

12. So or Topic 7.3.1 The Quadratic Formula Example 2 Solve 2x2 – 3x – 2 = 0 using the quadratic formula. Solution (continued)

13. Topic 7.3.1 The Quadratic Formula Example 4 Solve 2x2 – 11x + 13 = 0 using the quadratic formula. Solution Start by writing down the values of a, b, and c: a = 2, b = –11, and c = 13 Now very carefully substitute these into the quadratic formula. Then simplify to find the values of x. Solution continues… Solution follows…

14. So or Topic 7.3.1 The Quadratic Formula Example 4 Solve 2x2 – 11x + 13 = 0 using the quadratic formula. Solution (continued)

15. Topic 7.3.1 The Quadratic Formula Guided Practice Use the quadratic formula to solve each of the following equations. 1. x2 – 2x – 143 = 0 2. 2x2 + 3x – 1 = 0 3. x2 + 2x – 1 = 0 4. x2 + 3x + 1 = 0 5. 2x2 – 5x + 2 = 0 x = 13, –11 Solution follows…

16. Topic 7.3.1 The Quadratic Formula Guided Practice Use the quadratic formula to solve each of the following equations. 6. 3x2 – 2x – 3 = 0 7. 2x2 – 7x – 3 = 0 8. 6x2 – x – 1 = 0 9. 18x2 + 3x – 1 = 0 10. 4x2 – 5x + 1 = 0 Solution follows…

17. Topic 7.3.1 The Quadratic Formula Guided Practice 11. The equation 2x2 – 7x – 4 = 0 factors to (2x + 1)(x – 4) = 0. Using the zero product property we can find that x = – or x = 4. Verify this using the quadratic formula. 12. The height of a triangle is 4 ft more than 4 times its base length. If the triangle’s area is ft2, find the length of its base. Solution follows…

18. Topic 7.3.1 The Quadratic Formula Independent Practice Use the quadratic formula to solve each of the following equations. 1. 5x2 – 11x + 2 = 0 2. 2x2 + 7x + 3 = 0 3. 7x2 + 6x – 1 = 0 4. x2 – 7x + 5 = 0 5. 10x2 + 7x + 1 = 0 Solution follows…

19. Topic 7.3.1 The Quadratic Formula Independent Practice Use the quadratic formula to solve each of the following equations. 6. 3y2 – 8y – 3 = 0 7. 5x2 – 2x – 3 = 0 8. 4x2 + 3x – 5 = 0 9. 4t2 + 7t – 2 = 0 10. 6m2 + m – 1 = 0 Solution follows…

20. Topic 7.3.1 The Quadratic Formula Independent Practice Use the quadratic formula to solve each of the following equations. 11. 2x2 – x = 1 12. 3x2 – 5x = 2 13. 2x2 + 7x = 4 14. 4x2 + 17x = 15 15. 4x2 – 13x + 3 = 0 Solution follows…

21. Topic 7.3.1 The Quadratic Formula Independent Practice Use the quadratic formula to solve each of the following equations. 16. 4x2 – 1 = 0 17. 25x2 – 9 = 0 18. 4x2 + 15x = 4 19. 10x2 + 1 = 7x 20. 16x2 + 3 = 26x Solution follows…

22. Topic 7.3.1 The Quadratic Formula Independent Practice Solve these equations by factoring and using the zero product property, then verify the solutions by solving them with the quadratic formula. 21.x2 + 4x + 4 = 0 22. 4y2 – 9 = 0 23. x2 – x – 12 = 0 24. 2x2 – 3x – 9 = 0 25. 6x2 + 29x = 5 26. 7x2 + 41x = 6 x = –2 x = 4, –3 Solution follows…

23. Topic 7.3.1 The Quadratic Formula Independent Practice 27. The length of a rectangle is 20 cm more than 4 times its width. If the rectangle has an area of 75 cm2, find its dimensions. 28. The equation h = –14t2 + 12t + 2 gives the height of a tennis ball tseconds after being hit. How long will the ball take before it hits the ground? 2.5 cm by 30 cm 1 second Solution follows…

24. Topic 7.3.1 The Quadratic Formula Round Up The quadratic formula looks quite complicated, but don’t let that put you off. If you work through the derivation of the formula then you should see exactly why it contains all the elements it does.