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1. 1. QUADRATIC EQUATIONS MATH10 ALGEBRA Quadratic Equations (Algebra and Trigonometry, Young 2nd Edition, page 113-135)
2. 2. Week 3 Day 3 GENERAL OBJECTIVE At the end of the chapter the students are expected to: • • • • Solve quadratic equations using different methods, Solve equations in quadratic form, Solve equations leading to quadratic equation, and Solve real-world problems that involve quadratic equation.
3. 3. TODAY’S OBJECTIVE Week 3 Day 3 At the end of the lesson the students are expected to: • To distinguish between pure quadratic equation and complete quadratic equation, • To determine the number and type of solutions or roots of a quadratic equation based on the discriminant, • To define complex numbers, and • To solve quadratic equations by factoring, square root method, completing the square and quadratic formula.
4. 4. Week 3 Day 3 DEFINITION QUADRATIC EQUATION A quadratic equation in x is an equation that can be written in the standard form 2 ax  bx  c  0 where a, b, and c are real numbers and a  0 . a represents the numerical coefficient of x2 , b represents the numerical coefficient of x, and c represents the constant numerical term. Example: 2x2  0 2x2  7x  0 2x2  50  0 5x 2  3x  3  0
5. 5. Week 3 Day 3 • Pure Quadratic Equation If b=0, then the quadratic equation is termed a "pure" quadratic equation. Example: 3x2 +6=0 • Complete Quadratic Equation If the equation contains both an x and x2 term, then it is a "complete" quadratic equation. The numerical coefficient c may or may not be zero in a complete quadratic equation. Example: x2 +5x+6=0 and 2x2 - 5x = 0
6. 6. DEFINITION Week 3 Day 3 DISCRIMINANT OF A QUADRATIC EQUATION The term inside the radical, b2 -4ac, is called the discriminant. The discriminant gives important information about the ax 2  bx  c  0 corresponding solutions or roots of where a, b, and c are real numbers and a  0 . b2 -4ac Positive Zero Negative Solutions or Roots Two distinct real roots One real root (a double or repeated root) Two complex roots(complex conjugates)
7. 7. Week 3 Day 3 EXAMPLE Determine the nature of roots of the following quadratic equation. 1. x 2  4 x  5  0 2. x 2  2 x  5  0 3. 4 x 2  12 x  9  0 4. 9 x 2  12 x  4  0 5. x 2  6 x  3  0
8. 8. Week 3 Day 3 DEFINITION COMPLEX NUMBER A complex number is an expression of the form a  bi where a and b are real numbers and i   1  i2  1 a is the real part and b is the imaginary part . EXAMPLE Real Part Imaginary Part 3  4i 3 4 1 2  i 2 3 1 2 2  3 6i 0 6 -7 -7 0
9. 9. SOLVING QUADRATIC EQUATIONS Week 3 Day 3 There are four algebraic methods of solving quadratic equation in one variable, namely: • solution by factoring • solution by square root method • solution by completing the square • solution by quadratic formula
10. 10. Week 3 Day 3 SOLVING QUADRATIC EQUATIONS BY FACTORING The Factoring Method applies the Zero Product Property which states that if the product of two or more factors equals zero, then at least one of the factors equals zero. Thus if B·C=0, then B=0 or C=0 or both. STEPS: 1. Write the equation in standard form ax2 + bx + c = 0. 2. Factor the left side completely. 3. Apply the Zero Product Property to find the solution set.
11. 11. Week 3 Day 3 EXAMPLE Solve the following equations. 1. Classroom ex.1.3.1 2 x  12x  35  0 pp.114 Classroom ex. 1.3.2 2 2. 3t  10t  6  2 pp.115 Classroom ex. 1.3.3 3. 5y2  12y pp.95 6. x(2x  5)  3 7. (4x  1)(3x  1)  13 2 8.  4  x  3 x 4. #13 pp.124 9p2  12p  4 #16 5. 16v 2  25  0 page 124
12. 12. 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. Thus, if x 2  P, then x   P . NOTE : 1. The variable squared must be isolated first ( coefficien t equal to 1) 2. If P  0 is a real number , the equation x 2  P has real 2 distinct real solutions; x  P and x   P 2. If P  0, the equation x 2  P has a double root of zero 3. If P  0, the equation x 2  P has exactly two imaginary solutions
13. 13. Week 3 Day 3 EXAMPLE Solve the following equations. 1. Classroom ex.1.3.4 pp.116 2x  32  0 2 Classroom ex. 1.3.5 2. 5a2  10  0 pp.117 4. #30 pp.124 (4x - 1)2  16 #16 5. 16v 2  25  0 page 124 Classroom ex. 1.3.6 3. (x - 3)2  25 pp.117 6. 2  y 2  4 y  72    2 2 7. 2m  5 m  3   from College Algebra by Exconde, Marquez and Sabino exercise 5.4 numbers 9 and 10 page 112
14. 14. Week 3 Day 3 SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE STEPS: 1. Express the quadratic equation in the following form x 2  bx  c 2. Divide b by 2 and square the result, then add the square to both sides. 2 b b x  bx     c    2 2 2 2
15. 15. Week 3 Day 3 SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE 3. Write the left side of the equation as a perfect square 2  b b x    c     2 2 2 4. Solve using the square root method.
16. 16. Week 3 Day 3 EXAMPLE Solve the following equations. 1. Example #7 2 x  8x  3  0 pp.118 Example #2 2. 3x2  12x  13  0 pp.119 #55 x 2 1 3.  2x  pp.124 2 4 #56 4. pp.124 t2 2t 5   0 3 3 6 What number should be added to complete the square of each expression . #41 page 124 2 x2  x 5 #41 6. page 124 x 2 12x 5.
17. 17. 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:  b  b 2  4ac x 2a Note : The quadratic equation must be in standard form ax 2  bx  c  0 in order to identify the parameters a, b, c.
18. 18. DERIVATION OF QUADRATIC FORMULA BY COMPLETING THE SQUARE Consider the most general quadratic equation: ax2  bx  c  0 Solve by completing the square: WORDS MATH 1. Divide the equation by the leading coefficient a. 2. Subtract c from both sides. a b c x2  x   0 a a b c x  x a a 2 3. Subtract half of b and add the result  b     2a  sides. 4.. Write the 2 a to both left side of the equation as a perfect square and the right side as a single fraction. 2 2 b b b c x2  x        a  2a   2a  a 2 b  b2  4ac  x    4a2  2a 
19. 19. DERIVATION OF QUADRATIC FORMULA BY COMPLETING THE SQUARE WORDS 5. Solve using the square root method. b 6. Subtract from both sides 2a and simplify the radical. 7. Write as a single fraction. 8. We have derived the quadratic formula. MATH b b2  4ac x  2a 4a2 b b2  4ac x  2a 2a  b  b2  4ac x 2a
20. 20. Week 3 Day 3 EXAMPLE Solve the following equations using the quadratic formula. 1. Your Turn 2 x  2  2x pp.121 2. Example #11 4x 2  4 x  1  0 pp.121
21. 21. SUMMARY Week 3 Day 3  The four methods for solving quadratic equations are: 1. factoring 3. completing the square 2. square root method 4. quadratic formula  Factoring and the square root method are the quickest and easiest but cannot always be used.  Quadratic formula and completing the square work for all quadratic equations and can yield three types of solutions: 1. two distinct real roots 2.one real root (repeated) 3.or two complex roots (conjugates of each other)
22. 22. Week 4 Day 1 EQUATIONS IN QUADRATIC FORM (OTHER TYPES)
23. 23. Week 4 Day 1 CLASSWORK Solve each quadratic equation using any method. #87 2 2 4 1 1. t  t page 125 3 3 5 #89 12 2. x  7 page 125 x #91 4x - 2  3 -3 3.  page 125 x3 x x x - 3 #93 4. x2 0.1x  0.12 page 125
24. 24. TODAY’S OBJECTIVE Week 4 Day 1 At the end of the lesson the students are expected to: • To find the sum and product of roots of a quadratic equation. • To find the quadratic equation given the roots. • To transform a difficult equation into a simpler linear or quadratic equation, • To recognize the need to check solutions when the transformation process may produce extraneous solutions, • To solve radical equations.
25. 25. Week 4 Day 1 RECALL  The four methods for solving quadratic equations are: 1. factoring 3. completing the square 2. square root method 4. quadratic formula  Factoring and the square root method are the quickest and easiest but cannot always be used.  Quadratic formula and completing the square work for all quadratic equations and can yield three types of solutions: 1. two distinct real roots 2.one real root (repeated) 3.or two complex roots (conjugates of each other)
26. 26. SUM AND PRODUCT OF ROOTS Recall from the quadratic formula that when ax 2  bx  c  0   b  b2  4ac x 2a  b r   Let the roots be r and s   b  s   b 2  4ac 2a b 2  4ac 2a Week 4 Day 1
27. 27. SUM OF ROOTS Sum of roots = r + s  b  b 2  4ac  b  b 2  4ac rs   2a 2a  b  b 2  4ac  b  b 2  4ac  2a  2b rs  2a b rs   a Week 4 Day 1
28. 28. Week 4 Day 1 PRODUCT OF ROOTS Product of roots = (r) (s)  b  b 2  4ac  b  b 2  4ac (r)( s )  * 2a 2a  (  b)  2 b 2  4ac 4a 2 b 2  b 2  4ac (r )(s)  4a 2 c (r)(s)  a  2
29. 29. Week 4 Day 1 EXAMPLE Determine the value of k that satisfies the given condition 1. kx 2  5x  4  0; sum of roots is 20. 2. (3k  2)x 2  2x  k - 1  0; product of roots is - 1. 3. 2k  1x 2 - 10x  k 2  5k  6  0; one of the roots is 0. 4. 2k - 1x 2  k - 5 x - 6  0; the roots numerically equal but with opposite signs
30. 30. Week 4 Day 1 FINDING THE QUADRATIC EQUATION GIVEN THE ROOTS Let the roots be r and s, the quadratic equation is x - r x  s   0 Example: Find the quadratic equations with the given roots. 1. 3 1 and 4 2 2. 5 2 and  7 2 3. 1  2i and 1  2i
31. 31. RADICAL EQUATIONS Week 4 Day 1 Radical Equations are equations in which the variable is inside a radical (that is square root, cube root, or higher root). x  3  2, 2x  3  x, x  2  7x  2  6
32. 32. RADICAL EQUATIONS Week 4 Day 1 Steps in solving radical equations: 1. Isolate the term with a radical on one side. 2. Raise both (entire)sides of the equation to the power that will eliminate this radical and simplify the equation. 3. If a radical remains, repeat steps 1 and 2. 4. Solve the resulting linear or quadratic equation. 5. 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.
33. 33. EXAMPLE Week 4 Day 1 Solve the following equations. 1. Example 1 pp.128 x -3  2 Classroom ex. 1.4.1 a. 2. pp.128 b. Your Turn 3. pp.128 #29 6. page 133 2x  6  x  3 2 x  x Example 3 pp.129 x  2  7x  2  6 #28 5. page 133 4. x  5  1 x 2 #24 page 133 2x2  8x  1  x  3 2-x  3    2 - x  3  7.
34. 34. Week 4 Day 1 CATCH THE MISTAKE #83 8. Explain the mistake that is made. page 134 Solve the equation 3t  1  4 Solution : 3t  1  16 3t  15 t 5 This is incorrect.What mistake was made?
35. 35. Week 4 Day 1 SUMMARY .  b r   Let the roots be r and s   b  s   Sum of roots : rs   b a b 2  4ac 2a b 2  4ac 2a Productof roots : c (r)(s)  a Steps in solving radical equations: 1. Isolate the term with a radical on one side. 2. Raise both (entire)sides of the equation to the power that will eliminate this radical and simplify the equation. 3. If a radical remains, repeat steps 1 and 2. 4. Solve the resulting linear or quadratic equation. 5. Check the solutions and eliminate any extraneous solutions.
36. 36. TODAY’S OBJECTIVE Week 4 Day 2 At the end of the lesson the students are expected to: • To solve equations that are quadratic in form, • To realize that not all polynomial equations are factorable. • To solve equations that are factorable.
37. 37. Week 4 Day 2 EQUATIONS QUADRATIC IN FORM: u-SUBSTITUTION Equations that are higher order or that have fractional powers often can be transformed into quadratic equation by introducing a usubstitution, thus the equation is in quadratic form. Example: Original Equation Substitution New Equation x  3x  4  0 ux 2 u  3u  4  0 1 t3 u2  2u  1  0 4 2 2 t3 1  2t 3 1  0 u 2
38. 38. Week 4 Day 2 EQUATIONS QUADRATIC IN FORM: u-SUBSTITUTION Steps in solving equations quadratic in form: 1. Identify the substitution. 2. Transform the equation into a quadratic equation. 3. Apply the substitution to rewrite the solution in terms the original variable. 4. Solve the resulting equation. 5. Check the solution in the original equation.
39. 39. EXAMPLE Week 4 Day 2 Solve the following equations. 1. Example 4 -2 1 x  x  12  0 pp.131 Classroom ex. 1.4.4 2x  12  102x  11  9  0 2. pp.131 3. Classroom ex. 1.4.5 pp.131 2 z5 1  8z 5 8 5  16  0 2 Classroom ex. 1.4.5 3 4. 2x  x 3  x 3  0 pp.132
40. 40. Week 4 Day 2 FACTORABLE EQUATIONS  EQUATIONS WITH RATIONAL EXPONENTS BY FACTORING 7 4 1 Example #6 3 x  3x 3  4x 3  0 page 132 3 1 1  #71 2  5y 2  6y 2  0 y page 132  POLYNOMIAL EQUATION USING FACTORING BY GROUPING Classroom Ex. 1.4.7 6x3  12x 2  2x  4  0 page 132 #66 vv  33  40v  32  0 page 133