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• Week 3 Day 3
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• ### Quadratic equations lesson 3

1. 1. QUADRATIC EQUATIONS <br />MATH10 <br />ALGEBRA<br />Quadratic Equations (Algebra and Trigonometry, Young 2nd Edition, page 113-135) <br />
2. 2. Week 3 Day 3<br />GENERAL OBJECTIVE<br />At the end of the chapter the students are expected to:<br /><ul><li>Solve quadratic equations using different methods,
3. 3. Solve equations in quadratic form,
5. 5. Solve real-world problems that involve quadratic equation.</li></li></ul><li>Week 3 Day 3<br />TODAY’S OBJECTIVE<br />At the end of the lesson the students are expected to:<br /><ul><li> To distinguish between pure quadratic equation and complete quadratic equation,
6. 6. To determine the number and type of solutions or roots of a quadratic equation based on the discriminant,
7. 7. To define complex numbers, and
8. 8. To solve quadratic equations by factoring, square root method, completing the square and quadratic formula. </li></li></ul><li>Week 3 Day 3<br />DEFINITION<br />QUADRATIC EQUATION<br />A quadratic equation in x is an equation that can be written in the standard form <br />where a, b, and c are real numbers and a  0 .<br />a represents the numerical coefficient of x2 , <br />b represents the numerical coefficient of x, and <br />c represents the constant numerical term. <br />Example:<br />
9. 9. Week 3 Day 3<br /><ul><li>Pure Quadratic Equation</li></ul>If b=0, then the quadratic equation is termed a "pure" quadratic equation. <br /> Example: 3x2 +6=0 <br /><ul><li>Complete Quadratic Equation</li></ul>If the equation contains both an x and x2 term, then it is a "complete" quadratic equation. <br />The numerical coefficient c may or may not be zero in a complete quadratic equation. <br /> Example: x2 +5x+6=0 and 2x2 - 5x = 0 <br />
10. 10. Week 3 Day 3<br />DEFINITION<br />DISCRIMINANT OF A QUADRATIC EQUATION<br />The term inside the radical, b2 -4ac, is called the discriminant.<br />The discriminant gives important information about the corresponding solutions or roots of <br />where a, b, and c are real numbers and a  0 .<br />Positive<br />Two distinct real roots<br />One real root (a double or repeated root)<br />Zero<br />Negative<br />Two complex roots(complex conjugates)<br />
11. 11. Week 3 Day 3<br />EXAMPLE<br />Determine the nature of roots of the following <br />quadratic equation.<br />
12. 12. Week 3 Day 3<br />DEFINITION<br />COMPLEX NUMBER<br /> A complex number is an expression of the form<br /> where a and b are real numbers and <br /> a is the real part and b is the imaginary part .<br />EXAMPLE<br />3<br />4<br />0<br />6<br />-7<br />0<br />-7<br />
13. 13. Week 3 Day 3<br />SOLVING QUADRATIC EQUATIONS<br />There are four algebraic methods of solving quadratic equation <br />in one variable, namely:<br /><ul><li> solution by factoring
14. 14. solution by square root method
15. 15. solution by completing the square
16. 16. solution by quadratic formula</li></li></ul><li>Week 3 Day 3<br />SOLVING QUADRATIC EQUATIONS BY FACTORING<br />The Factoring Method applies the Zero Product Property whichstates that if the product of two ormore factors equals zero, then at least one of the factors equals zero. <br /> Thus if B·C=0, then B=0 or C=0 or both.<br />STEPS:<br />Write the equation in standard form ax2 + bx + c = 0.<br />Factor the left side completely.<br />Apply the Zero Product Property to find the solution set.<br />
17. 17. Week 3 Day 3<br />EXAMPLE<br />Solve the following equations.<br />
18. 18. Week 3 Day 3<br />SOLVING QUADRATIC EQUATIONS BY SQUARE ROOT METHOD<br />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.<br />
19. 19. Week 3 Day 3<br />EXAMPLE<br />Solve the following equations.<br />
20. 20. Week 3 Day 3<br />SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE<br />STEPS:<br />
21. 21. Week 3 Day 3<br />SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE<br />
22. 22. Week 3 Day 3<br />EXAMPLE<br />Solve the following equations.<br />
23. 23. Week 3 Day 3<br />SOLVING QUADRATIC EQUATIONS BY QUADRATIC FORMULA<br />THE QUADRATIC FORMULA<br />The roots of the quadratic equation ax2 + bx + c = 0, where a, b, and c are constants and a  0 are given by:<br /> <br /> <br /> <br />
24. 24. DERIVATION OF QUADRATIC FORMULA BY COMPLETING THE SQUARE<br />Consider the most general quadratic equation: <br />Solve by completing the square: <br />Divide the equation by the leading coefficient a.<br />2. Subtract from both sides.<br />3. Subtract half of and add<br /> the result to both sides.<br />4. Write the left side of the equation as a perfect square and the right side as a single fraction.<br />
25. 25. DERIVATION OF QUADRATIC FORMULA BY COMPLETING THE SQUARE<br />Solve using the square root method.<br />6. Subtract from both sides<br /> and simplify the radical.<br />7. Write as a single fraction.<br />8. We have derived the quadratic formula.<br />
26. 26. Week 3 Day 3<br />EXAMPLE<br />Solve the following equations using the quadratic formula.<br />
27. 27. Week 3 Day 3<br />SUMMARY <br /><ul><li>The four methods for solving quadratic equations are: </li></ul> 3. completing the square<br />1. factoring <br />2. square root method<br />4. quadratic formula<br /><ul><li>Factoring and the square root method are the quickest and easiest but cannot always be used.
28. 28. Quadratic formula and completing the square work for all </li></ul> quadratic equations and can yield three types of solutions:<br /> two distinct real roots<br />one real root (repeated)<br />or two complex roots (conjugates of each other)<br />
29. 29. Week 4 Day 1<br />EQUATIONS IN QUADRATIC FORM<br />(OTHER TYPES) <br />
30. 30. Week 4 Day 1<br />CLASSWORK<br />
31. 31. TODAY’S OBJECTIVE<br />Week 4 Day 1<br />At the end of the lesson the students are expected to:<br /><ul><li>To find the sum and product of roots of a quadratic equation.
32. 32. To find the quadratic equation given the roots.
33. 33. To transform a difficult equation into a simpler linear or quadratic equation,
34. 34. To recognize the need to check solutions when the transformation process may produce extraneous solutions,
35. 35. To solve radical equations.</li></li></ul><li>Week 4 Day 1<br />RECALL <br /><ul><li>The four methods for solving quadratic equations are: </li></ul> 3. completing the square<br />1. factoring <br />2. square root method<br />4. quadratic formula<br /><ul><li>Factoring and the square root method are the quickest and easiest but cannot always be used.
36. 36. Quadratic formula and completing the square work for all </li></ul> quadratic equations and can yield three types of solutions:<br /> two distinct real roots<br />one real root (repeated)<br />or two complex roots (conjugates of each other)<br />
37. 37. SUM AND PRODUCT OF ROOTS<br /> Recall from the quadratic formula that when <br />Week 4 Day 1<br />
38. 38. Week 4 Day 1<br />SUM OF ROOTS<br />Sum of roots = r + s<br />
39. 39. Week 4 Day 1<br />PRODUCT OF ROOTS<br />Product of roots = (r) (s)<br />
40. 40. Week 4 Day 1<br />EXAMPLE<br />Determine the value of k that satisfies the given condition<br />
41. 41. Week 4 Day 1<br />FINDING THE QUADRATIC EQUATION GIVEN THE ROOTS <br />Example: Find the quadratic equations with the given roots.<br />
42. 42. Week 4 Day 1<br />RADICAL EQUATIONS<br />Radical Equations are equations in which the variable is<br />inside aradical (that is square root, cube root, or higher <br />root). <br />
43. 43. Week 4 Day 1<br />RADICAL EQUATIONS<br />Steps in solving radical equations:<br />Isolate the term with a radical on one side.<br />Raise both (entire)sides of the equation to the power that will eliminate this radical and simplify the equation.<br />If a radical remains, repeat steps 1 and 2.<br />Solve the resulting linear or quadratic equation.<br />Check the solutions and eliminate any extraneous solutions.<br />Note: When both sides of the equations are squared extraneous solutions can arise , thus checking is part of the solution. <br />
44. 44. EXAMPLE<br />Week 4 Day 1<br />Solve the following equations.<br />
45. 45. Week 4 Day 1<br />CATCH THE MISTAKE<br />
46. 46. SUMMARY <br />Week 4 Day 1<br /><ul><li>.
47. 47. Steps in solving radical equations:</li></ul>Isolate the term with a radical on one side.<br />Raise both (entire)sides of the equation to the power that will eliminate this radical and simplify the equation.<br />If a radical remains, repeat steps 1 and 2.<br />Solve the resulting linear or quadratic equation.<br />Check the solutions and eliminate any extraneous solutions.<br />
48. 48. TODAY’S OBJECTIVE<br />Week 4 Day 2<br />At the end of the lesson the students are expected to:<br /><ul><li>To solve equations that are quadratic in form,
49. 49. To realize that not all polynomial equations are factorable.
50. 50. To solve equations that are factorable.</li></li></ul><li>Week 4 Day 2<br />EQUATIONS QUADRATIC IN FORM: u-SUBSTITUTION<br />Equations that are higher order or that have fractional powers often can be transformed into quadratic equation by introducing a u-substitution, thus the equation is in quadratic form.<br />Example:<br />
51. 51. Week 4 Day 2<br />EQUATIONS QUADRATIC IN FORM: u-SUBSTITUTION<br />Steps in solving equations quadratic in form:<br />Identify the substitution.<br />Transform the equation into a quadratic equation.<br />Apply the substitution to rewrite the solution in terms the original variable.<br />Solve the resulting equation.<br />Check the solution in the original equation.<br />
52. 52. EXAMPLE<br />Week 4 Day 2<br />Solve the following equations.<br />
53. 53. Week 4 Day 2<br />FACTORABLE EQUATIONS <br /><ul><li>EQUATIONS WITH RATIONAL EXPONENTS BY FACTORING