10. Week 3 Day 3 DEFINITION DISCRIMINANT OF A QUADRATIC EQUATION The term inside the radical, b2 -4ac, is called the discriminant. The discriminant gives important information about the corresponding solutions or roots of where a, b, and c are real numbers and a 0 . Positive Two distinct real roots One real root (a double or repeated root) Zero Negative Two complex roots(complex conjugates)
11. Week 3 Day 3 EXAMPLE Determine the nature of roots of the following quadratic equation.
12. Week 3 Day 3 DEFINITION COMPLEX NUMBER A complex number is an expression of the form where a and b are real numbers and a is the real part and b is the imaginary part . EXAMPLE 3 4 0 6 -7 0 -7
17. Week 3 Day 3 EXAMPLE Solve the following equations.
18. Week 3 Day 3 SOLVING QUADRATIC EQUATIONS BY SQUARE ROOT METHOD The Square Root Property states that if an expression squared is equal to a constant , then the expression is equal to the positive or negative square root of the constant.
19. Week 3 Day 3 EXAMPLE Solve the following equations.
20. Week 3 Day 3 SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE STEPS:
21. Week 3 Day 3 SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE
22. Week 3 Day 3 EXAMPLE Solve the following equations.
23. Week 3 Day 3 SOLVING QUADRATIC EQUATIONS BY QUADRATIC FORMULA THE QUADRATIC FORMULA The roots of the quadratic equation ax2 + bx + c = 0, where a, b, and c are constants and a 0 are given by:
24. DERIVATION OF QUADRATIC FORMULA BY COMPLETING THE SQUARE Consider the most general quadratic equation: Solve by completing the square: Divide the equation by the leading coefficient a. 2. Subtract from both sides. 3. Subtract half of and add the result to both sides. 4. Write the left side of the equation as a perfect square and the right side as a single fraction.
25. DERIVATION OF QUADRATIC FORMULA BY COMPLETING THE SQUARE Solve using the square root method. 6. Subtract from both sides and simplify the radical. 7. Write as a single fraction. 8. We have derived the quadratic formula.
26. Week 3 Day 3 EXAMPLE Solve the following equations using the quadratic formula.
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28. Quadratic formula and completing the square work for all quadratic equations and can yield three types of solutions: two distinct real roots one real root (repeated) or two complex roots (conjugates of each other)
29. Week 4 Day 1 EQUATIONS IN QUADRATIC FORM (OTHER TYPES)
32. To find the quadratic equation given the roots.
33. To transform a difficult equation into a simpler linear or quadratic equation,
34. To recognize the need to check solutions when the transformation process may produce extraneous solutions,
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36. Quadratic formula and completing the square work for all quadratic equations and can yield three types of solutions: two distinct real roots one real root (repeated) or two complex roots (conjugates of each other)
37. SUM AND PRODUCT OF ROOTS Recall from the quadratic formula that when Week 4 Day 1
38. Week 4 Day 1 SUM OF ROOTS Sum of roots = r + s
39. Week 4 Day 1 PRODUCT OF ROOTS Product of roots = (r) (s)
40. Week 4 Day 1 EXAMPLE Determine the value of k that satisfies the given condition
41. Week 4 Day 1 FINDING THE QUADRATIC EQUATION GIVEN THE ROOTS Example: Find the quadratic equations with the given roots.
42. Week 4 Day 1 RADICAL EQUATIONS Radical Equations are equations in which the variable is inside aradical (that is square root, cube root, or higher root).
43. Week 4 Day 1 RADICAL EQUATIONS Steps in solving radical equations: Isolate the term with a radical on one side. Raise both (entire)sides of the equation to the power that will eliminate this radical and simplify the equation. If a radical remains, repeat steps 1 and 2. Solve the resulting linear or quadratic equation. Check the solutions and eliminate any extraneous solutions. Note: When both sides of the equations are squared extraneous solutions can arise , thus checking is part of the solution.
47. Steps in solving radical equations:Isolate the term with a radical on one side. Raise both (entire)sides of the equation to the power that will eliminate this radical and simplify the equation. If a radical remains, repeat steps 1 and 2. Solve the resulting linear or quadratic equation. Check the solutions and eliminate any extraneous solutions.
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49. To realize that not all polynomial equations are factorable.
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51. Week 4 Day 2 EQUATIONS QUADRATIC IN FORM: u-SUBSTITUTION Steps in solving equations quadratic in form: Identify the substitution. Transform the equation into a quadratic equation. Apply the substitution to rewrite the solution in terms the original variable. Solve the resulting equation. Check the solution in the original equation.
56. Week 4 Day 3 Start RECALL A Read and analyze the problem Make a diagram or sketch if possible Solve the equation Determine the unknown quantity. Check the solution Set up an equation, assign variables to represent what you are asked to find. no Is the unknown solved? no yes yes Did you set up the equation? A End
57. Week 4 Day 3 APPLICATION PROBLEMS 1. If a person drops a water balloon off the rooftop of an 81 foot building, the height of the water balloon is given by the equation where t is in seconds. When will the water balloon hit the ground? (Classroom example 1.3.12 page 122) You have a rectangular box in which you can place a 7 foot long fishing rod perfectly on the diagonal. If the length of the box is 6 feet, how wide is that box? (Classroom example 1.3.13 page 123) 3. A base ball diamond is a square. The distance from base to base is 90 feet. What is the distance from the home plate to the second base? (#108 page 125)
58. Week 4 Day 3 4. Lindsay andKimmie, working together, can balance the financials for the Kappa Kappa Gama sorority in 6days. Lindsay by herself can complete the job in 5days less than Kimmie. How long will it take Lindsay to complete the job by herself? (# 113 page 125) 5.A rectangular piece of cardboard whose length is twice its width is used to construct an open box. Cutting a I foot by 1 foot square off of each corner and folding up the edges will yield an open box. If the desired volume is 12 cubic feet, what are the dimensions of the original piece of cardboard? (# 110 page 125) 6.Aspeed boat takes 1 hour longer to go 24 miles up a river than to return. If the boat cruises at 10mph in still water, what is the rate of the current? (#140 page 126)
59. Week 4 Day 3 7.Cost for health insurance with a private policy is given by where C is the cost per day and a is the insured’s age in years. Health insurance for a six year old, a=6, is $4 a day (or $1,460 per year). At what age would someone be paying $9 a day (or $3,285 per year). (#73 page 134) 8. The period (T) of a pendulum is related to the length (L) of the pendulum and acceleration due to gravity (g) by the formula . If the gravity is and the period is 1 second find the approximate length of the pendulum. Round to the nearest inch. (#80 page 134)
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