Algebra II Unit #2 4.6 NOTES: Solving Quadratic Equations (More Methods) Block:

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1 Algebra II Unit # Name: 4.6 NOTES: Solving Quadratic Equations (More Methods) Block: (A) Background Skills - Simplifying Radicals To simplify a radical that is not a perfect square: (B) Solving Simple Square Roots: x = 9 m = 40 q = 108 **Important note: When you solve a square root there are solutions: ( and ) (C) METHOD #3: Finding solutions using SQUARE ROOTS. Give your exact answer (no rounding). **Important things to remember: 1. Work backward to isolate the x or (x ± #) term. When you square root both sides, set the term equal to both POSITIVE and NEGATIVE value 5x 405 x x 40 ( x 4) x 4 x Practice: Solve each equation using square roots. Show all steps of your work and write the exact answer. (leave in simplified radical form, not a decimal). 1) r 96 r 8 x 50

2 ) 4x m x ) 10 x p 4 16 k 1 9 4) x m m Converting standard form to vertex form by COMPLETING THE SQUARE 1. Convert the expressions into standard form. Show all steps of the distribution! a) x b) x 5 c) x 9 d) x 7 The expressions above are called SQUARED because they have terms. When we put them in standard form, they are then called because they have terms. We specifically call them.

3 . Suppose we are given a perfect square trinomial, but do not know the last term (c). How can we find it? a) x 8x b) x 0x c) x 10x d) x 1x Practice: Find the missing value (b or c) that would complete the square. e) x + 18x f) x 14x g) x + x + 49 h) x x + 11 **Vertex Form f(x) = a(x h) + k looks like a "squared binomial" with an added k value. Therefore, it's helpful to be able to write a standard quadratic equation as a perfect square trinomial or squared binomial. **Important things to remember: You can create a perfect square trinomial by completing the square. 1. Write the equation in standard form:. Isolate the constant 3. Make the equation so that a = (By dividing everything by a ) 4. TO COMPLETE THE SQUARE: add the "perfect c value" to the equation ON BOTH SIDES 5. Factor the trinomial into a binomial = a constant value. 6(a). Simplify to form. 6(b). Solve for x **If solving this equation, you could use the method from this point!. We can use this tool / strategy to turn standard form quadratic functions y = ax + bx + c into vertex form quadratic functions y = a(x h) + k. Why would we do this?? 3. Write the following quadratic function in vertex form by completing the square. Identify the key features. a) y x 10x Describe ALL transformations of the parent function f(x) = x What are the coordinates of the vertex? What are the coordinates of the y-intercept? 3

4 b) y x 18x 95 Describe ALL transformations of the parent function f(x) = x What are the coordinates of the vertex? What are the coordinates of the y-intercept? c) y 5x 0x 11 Describe ALL transformations of the parent function f(x) = x What are the coordinates of the vertex? What are the coordinates of the y-intercept? d) y x 1x 14 Describe ALL transformations of the parent function f(x) = x What are the coordinates of the vertex? What are the coordinates of the y-intercept? e) y 4x 16x 7 Describe ALL transformations of the parent function f(x) = x What are the coordinates of the vertex? What are the coordinates of the y-intercept? 4

5 SOLVING by Completing the Square: Solve by completing the square and using square roots to solve. Show all steps of your work. Add the "perfect c value" to both sides of the equation to keep it balanced! 1. x 10x 1. x 6x x 10x b 8b 5. d d x 1x 9 7. x 6x x 4x q 14q 5

6 4.7 Solving Equations with the Quadratic Formula (METHOD #4) To solve a quadratic equation in standard form set equal to 0... x "Pop Goes the Weasel" Song... What makes a Quadratic Equation Quadratic? Solve using the Quadratic Formula. Show your steps and give an exact answer as a simplified radical. Example 1 : 4x x 7 Example : x 5x 15 Use the quadratic formula to solve the equation. Show all steps of your work! Leave your answer exact as a simplified radical. Then give a decimal approximation for each solution to the tenths place = x 4x 5. x x = 6

7 3. 0 = 8x 8x x 8x + 1 = 0 5. x 5 = 3x 6. 3x 1x = 1 7. x = 3x x 5x = x 9. 5x + 40x = x 7x 3 7

8 *Key Concept: The Discriminant: In the quadratic formula, the expression is called the DISCRIMINANT of the associated equation in standard form f(x) = ax + bx + c. You can use the discriminant of a quadratic equation to determine the equation s and of solutions. **Using the Discriminant of ax bx c 0 ** Value of discriminant if b - 4ac > 0 if b - 4ac = 0 if b - 4ac < 0 Number and type of solutions y Graph of ax bx c For the following quadratic equations: a. Fill in the quadratic formula e. Sketch a graph of the parabola b. Highlight the discriminant f. Label the x-intercepts (if any) c. Find the value of the discriminant d. Determine the type and number of solutions Example 1: Example : x 4x 4 0 3x 1x 1 0 8

9 Example 3: For what values of c would the equation x 6x c 0 have imaginary solutions? Example 4: For what values of c would the equation x 16x c 0 have real solutions? Example 5: For what value of c would the equation x 9x c 0 have a discriminant of 113? Example 6: For what value of c would the equation 8x x c 0 have 1 real solution? Real-Life Application: You hit a golf ball in the air from a height of 1 inch about the ground with an initial velocity of 85 feet per second. The function h(t) = 16t + 85t + 1 models the height, in feet, of the ball at time, t, in seconds. 1 a. Will the ball reach a height of 115 feet? If so, when will this happen? b. Will the ball reach a height of 110 feet? If so, when will this happen? 9

10 Practice: Find the discriminant and determine the number and type of solutions. Find the discriminant of the quadratic equation and give the number and type of solutions of the equation. 1. x 8x + 16 = 0. 8x + 8x + 3 = 0 3. x + 7x + 11 = x + 16x = 11x 3x 5. 3x + 4x = 1x x 5 = x + 9x 7. For what values of c would the equation x + 8x + c = 0 have two real solutions? 8. For what values of c would the equation 3x + 4x + c = 0 have two imaginary solutions? 9. For what value of c would the equation x x + c = 0 have 1 real solution? 10. What is the value of c if the discriminant of x + 5x + c = 0 is 3? 10