The Quadratic Formula

Name: Algebra 2 Date:

The Quadratic Formula The trouble with finding zeros of quadratic functions by factoring is that it’s sometimes hard, or even impossible, to find the factorization. We have also seen functions that are quite difficult to find the zeroes for by completing the square. There is another algebraic method for finding zeros that will always succeed. That method involves using a formula for the zeros, called the Quadratic Formula. The formula says that the zeros of the quadratic function f(x) = ax2 + bx + c are given by - b ± b 2 - 4ac x = 2a As the large type suggests, this is an extremely valuable formula. Memorize it. Use it. Love it. It’s one of the two most important formulas in the whole course. And once you know it, you’ll never be stuck unable to find zeros because the factoring was too hard. Example: Find the zeros of f(x) = 8x2 – 10x + 2.

- (-10) ± (-10) 2 - 4 ×8× 2 10 ± 100 - 64 10 ± 36 10 ± 6 x = = = = . 2 ×8 16 16 16 10 + 6 16 10 - 6 4 1 So the two zeros are: = = 1 and = = . 16 16 16 16 4 Example: Find the zeros of x2 + 5x + 3. - 5 ± 52 - 4 ×1×3 - 5 ± 13 x = = 2 ×1 2 - 5 + 13 - 5 - 13 So the two zeros are: » -0.697 and » -4.303 2 2

Think about, then discuss: How many zeros? As you can see in the examples above, the Quadratic Formula can produce two answers, due to the ± (“plus or minus”) in the formula. But there won’t always be two zeros, because different things can happen in the part of the formula underneath the . For example, in the number system we’re using, negative numbers don’t have square roots at all. Think about the different possibilities, then fill in these sentences.

 f(x) = ax2 + bx + c will have two solutions when ______.

 f(x) = ax2 + bx + c will have one solution when ______.

 f(x) = ax2 + bx + c will have no real solutions when ______. Name: Algebra 2 Date:

Problem Set 1. Fill in the Quadratic Formula below. OK to just copy it from the front page. - ± x =

2. a. Find the zeros of f(x) = 2x2 + 7x + 3 using the Quadratic Formula.

b. Check your answer by finding the zeros by a second method: factoring. Make sure your answer agrees with part a.

3. a. Find the zeros of f(x) = 4x2 – 8x – 21 using the Quadratic Formula.

b. Check your answer by finding the zeros by a second method: factoring. Make sure your answer agrees with part a. Name: Algebra 2 Date:

4. Fill in the Quadratic Formula again: ± x =

5. a. Find the zeros of f(x) = 3x2 + 5x – 2 using the Quadratic Formula.

b. Check your answer by finding the zeros graphically (using [2nd][TRACE]zero on your calculator). Make sure your answer agrees with part a.

6. a. Find the zeros of f(x) = –2x2 + 3x + 4 using the Quadratic Formula.

b. Check your answer by finding the zeros graphically (using [2nd][TRACE]zero on your calculator). Make sure your answer agrees with part a. Name: Algebra 2 Date:

7. Fill in the Quadratic Formula again. Try to do it without looking back this time. - ± - x =

8. a. Find the zero(s) of f(x) = 2x2 + 3x + 4 using the Quadratic Formula. Something different happens this time.

b. Confirm your answer by looking at the graph of f(x) = 2x2 + 3x + 4 on your calculator. Sketch the graph and explain what it tells you about zeros.

9. a. Find the zero(s) of f(x) = 4x2 + 12x + 9 using the Quadratic Formula.

b. Confirm your answer by looking at the graph of f(x) = 4x2 + 12x + 9 on your calculator. Sketch the graph and explain what it tells you about zeros. Name: Algebra 2 Date:

10. Solve these equations using the Quadratic Formula. Hint: First use addition and/or subtraction to get 0 on one side of the equation. Then use the Quadratic Formula. a. x2 + 3 = –4x

b. –2x2 – 12x = 18

c. 4x2 = 36x

d. x2 = 7 Name: Algebra 2 Date:

11. One more time: write the Quadratic Formula. Try to do it without looking back.

12. A ball is thrown out a window at time t = 0. After t seconds, the height of the ball is: h(t) = –16t2 + 30t + 12 feet. a. How high is the window from which the ball was thrown?

b. Using the Quadratic Formula, find the time t that the ball hits the ground.

13. A pumpkin is dropped to the ground from the roof of a tall building. Its height after t seconds is: h(t) = –4.9t2 + 50 meters. How far does the pumpkin fall, and how long does the fall take?