1. 1 Topic 7.3.1 The Quadratic Formula

2. 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. 3 Topic 7.3.1 The Quadratic Formula You can also use the quadratic formula to solve quadratic equations. It works every time.

4. 4 Topic 7.3.1 Quadratic Equations can be in Any Variable The Quadratic Formula The standard form for a quadratic equation is: ax 2 + bx + c = 0 ( a  0) 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. 5 Topic 7.3.1 Example 1 Solution follows… The Quadratic Formula Solution continues… a) Here, a = 1, b = –4, and c = 3. So using the zero property: ( x – 3) = 0 or ( x – 1) = 0, or x = 3 or x = 1 This equation factors to give ( x – 3)( x – 1) = 0 Solution Find the solutions of the following quadratic equations: a) x 2 – 4 x + 3 = 0 b) y 2 – 4 y + 3 = 0

6. 6 Topic 7.3.1 Example 1 The Quadratic Formula Solution (continued) Find the solutions of the following quadratic equations: a) x 2 – 4 x + 3 = 0 b) y 2 – 4 y + 3 = 0 b) Here, the variable is y rather than x — but that does not affect the solutions. Again a = 1, b = –4, and c = 3, so it is the same equation as in a), and will have the same solutions: y = 3 or y = 1 You can see that the two quadratic equations are really the same — only the variables have changed.

7. 7 Topic 7.3.1 The Quadratic Formula Solves Any Quadratic Equation The Quadratic Formula The solutions to the quadratic equation ax 2 + bx + c = 0 are given by the quadratic formula: You can derive the formula by completing the square.

8. 8 Topic 7.3.1 The Quadratic Formula ax 2 + bx + c = 0 ax 2 + bx = – c ax 2 + x = – 3. Divide the equation by a 1. Start with the standard form2. Subtract c 4. Complete the square 5. Add the fractions 7. Include positive and negative roots 6. Simplify 8. Rearrange to give the quadratic formula

9. 9 Topic 7.3.1 Example 2 Solution follows… The Quadratic Formula Solution Solution continues… Solve x 2 – 5 x – 14 = 0 using the quadratic formula. 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.

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

11. 11 Topic 7.3.1 Example 3 Solution follows… The Quadratic Formula Solution Solution continues… Solve 2 x 2 – 3 x – 2 = 0 using the quadratic formula. 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.

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

13. 13 Topic 7.3.1 Example 4 Solution follows… The Quadratic Formula Solution Solution continues… Solve 2 x 2 – 11 x + 13 = 0 using the quadratic formula. 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.

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

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

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

17. 17 Topic 7.3.1 Guided Practice Solution follows… The Quadratic Formula 11. The equation 2 x 2 – 7 x – 4 = 0 factors to (2 x + 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 ft 2, find the length of its base.

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

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

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

21. 21 Topic 7.3.1 Independent Practice Solution follows… The Quadratic Formula Use the quadratic formula to solve each of the following equations. 16. 4 x 2 – 1 = 0 17. 25 x 2 – 9 = 0 18. 4 x 2 + 15 x = 4 19. 10 x 2 + 1 = 7 x 20. 16 x 2 + 3 = 26 x

22. 22 Topic 7.3.1 Independent Practice Solution follows… The Quadratic Formula Solve these equations by factoring and using the zero product property, then verify the solutions by solving them with the quadratic formula. 21. x 2 + 4 x + 4 = 022. 4 y 2 – 9 = 0 23. x 2 – x – 12 = 024. 2 x 2 – 3 x – 9 = 0 25. 6 x 2 + 29 x = 526. 7 x 2 + 41 x = 6 x = –2 x = 4, –3

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

24. 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.