Accel Alg E. L. E. Notes Solving Quadratic Equations. Warm-up

Transcription

1 Accel Alg E. L. E. Notes Solving Quadratic Equations Warm-up Solve for x. Factor x 36 = 0 2. x 2 8x Factor. Factor. 3. 2x 2 + 5x 7 4. x 2 121

2 Solving Quadratic Equations Methods: (1. By Inspection) 2. Factoring 3. Completing the square 4. Quadratic Formula 5. Graphing 1, 2 and 5 are the simplest, but do not always work. 3 and 4 require you to memorize a formula, but they always work. Keep in mind, these will have two answers, or no real solutions.

3 1. Solving by INSPECTION. These are the problems without the linear term, ax 2 + c = 0. Either you almost see the answer right away, or it is very easy to isolate the variable and solve. Either you see it right away (and no work, but don t forget both answers), or 1. Isolate the quadratic term (i. e. get x 2 by itself) 2. Take the square root of both sides. Examples Solve. 1. x 2 = x 2 12 = x 2 12 = 0 x 2 = ± x = ±4 x 2 = 12 3x 2 = 12 x 2 = ± x = ±2 3 x 2 = 4 x 2 = ± 4 x = ±2

4 2. Solve by FACTORING. 1. We are using the zero product property to solve with this method. 2. Manipulate equation so that one side is Factor the other side. 4. Set all factors equal to zero and solve (find a value that works). 5. Substitute them back in to check your solutions. Examples Solve. 1. (x 2)(x + 3) = 0 2. x 2 4x = 0

5 3. x 2 4x = 12

6 Try these: Solve. 1. 5x 2 21x + 4 = 0 2. x 2 12x + 36 = x 2 6x 3 = 1 3x 4. 2x = 2 + 5x 5. x 2 + 6x + 5 = 0

7 3. Solve by COMPLETING THE SQUARE. 1. Manipulate the equation so that the quadratic and linear terms are one side, and everything else is on the other. 2. (If necessary) factor out a from the quadratic term and linear term. 3. Make room for a constant term that will be added to both sides of the equation. 4. With the quadratic side in the form x 2 + bx + a. divide b by 2, and square that value b. add that to both sides of the equation (goes in the blank) c. if a had to be factored out, then multiply a when adding to other side d. factor the perfect square trinomial (should require no work) 5. Use algebra skills you have learned to solve. Examples Solve. 1. x 2 + 6x + 5 = 0

8 2. x 2 + 4x 1 = 0

9 3. 3x 2 6x 9 = 0

10 Try these: Solve. 1. 3x 2 4x 9 = x 2 + 6x + 2 = x 2 + 5x = 2

11 4. Solve using the QUADRATIC FORMULA. This is a must have for all math courses from here on out!!! They will give you the formula on the EOC, but you need to go ahead and begin memorizing it for the future!!! Like, factoring, you must have one side of the equation be equal to 0 before you can use this method. If the quadratic is written in the form: ax 2 + bx + c = 0, then the zeros/roots/solutions can be found by: x = b ± b2 4ac 2a The Quadratic Formula Proof that this works comes from completing the square with equation in standard form. ax 2 + bx + c = 0

12 Examples Solve. x = b± b2 4ac 2a 1. x 2 + 6x + 5 = 0 x = b± b2 4ac 2a 2. x x 19 = 0

13 3. 3x 2 21x = 5 x = b± b2 4ac 2a

14 Try these. Solve. x = b± b2 4ac 2a 1. x 2 12x + 2 = 0 2. x x 8 = 0 3. x 2 + 7x = x 2 10x = 7

15 5. Solve by GRAPHING. This one is usually done with technology (graphing calculator). However, you can at least approximate the solutions if the parabola crosses the x-axis. The solutions to these equations are also referred to as the roots, zeros or x-intercepts. So, solving ax 2 + bx + c = 0 is the same as finding the x-intercepts of f(x) = ax 2 + bx + c or y = ax 2 + bx + c. So, to solve ax 2 + bx + c = 0: 1. Pretend that it is the function, f(x) = ax 2 + bx + c 2. Graph the parabola 3. Find where the parabola crosses the x-axis Examples Sovle. 1. x 2 + 6x + 5 = 0 Graph: y = x 2 + 6x + 5 Where does the graph cross the x-axis?

16 With the calculator (TI-83, TI-84, etc.): Turn the calculator ON Press Y = Type in your function into Y1 = Hit GRAPH or ZOOM then 6 You should see the parabola graphed, and you should at least be able to guess at the zeros. You may need to adjust the WINDOW to fit some functions that are not easily seen on a standard screen. To locate zeros (roots, x-intercepts), you can press 2nd TRACE. Then select zero or press 2 It will ask for a Left Bound? Use the arrows to move the cursor to the left of the zero and hit ENTER, or type in a value of x that you know is to the left of the zero and hit ENTER Next it will ask for a Right Bound? Use the arrows to move the cursor to the right of the zero and hit ENTER, or type in a value of x that you know is to the right of the zero and hit ENTER. Then it will ask for a Guess? You can usually just hit ENTER, or you may need to move the cursor close to the zero you are searching for, and then hit ENTER. Repeat the process to find the other zero.

17 Examples Solve the equations graphically. 1. 2x 2 + 5x 3 = x 2 x + 8 = 3x 2 6x 6

18 3. x 2 6x 8 = 0

19 Try these. Solve, graphically. 1. x 2 6x = 0 2. x 2 + 6x + 8 = 0 3. x 2 5 = 0 4. x 2 + 9x 4 = 0 5. x 2 + 2x+= 4

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