Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
FORMS OF QUADRATIC EQUATIONS TOPIC PAGE General form to transformational form (when a = 1) 4 General form to transformational form 8 (when a = 1) Transformational form to standard form 14 Standard form to transformational form 18 Standard form to general form 22 © 2003 Paul Lobel Careerwise Tutoring Easy Math Steps 722-3733 1 YOU SHOULD KNOW GENERAL FORM OF A QUADRATIC EQUATION y ax bx c 2 ANY NUMBER ANY NUMBER EXCEPT ZERO © 2003 Paul Lobel Careerwise Tutoring ANY NUMBER General form y 2x 2 4x 13 Standard form y 2(x 2)2 - 15 Transformational form 1 y 15 x 22 2 Mapping rule © 2003 Paul Lobel Careerwise Tutoring (x, y) - - - - - - - -- (x - 2, 2y - 15) 3 © 2003 Paul Lobel Careerwise Tutoring 4 EXAMPLE CHANGING FROM TRANSFORMATIONAL FORM TO STANDARD FORM Quadratic equation in transformational form 1 2 y 2 x 1 3 © 2003 Paul Lobel Careerwise Tutoring STEP ONE STEP TWO Change the number in front of the left Change the number after the bracket to its reciprocal and move it to y value to its opposite and move it the front of the right bracket to the right hand side. y 2 - 3x 1 2 y - 3x 1 - 2 2 5 PRACTICE QUESTIONS CHANGING FROM TRANSFORMATIONAL FORM TO STANDARD FORM Quadratic equation in transformational form STEP ONE STEP TWO Change the number in front of the left Change the number after the bracket to its reciprocal and move it to y value to its opposite and move it the front of the right bracket to the right hand side. 2y 2 x 1 2 2y 3 x 12 2 2 y 7 x 82 3 4y 4 x 4 2 1 © 2003 Paul Lobel Careerwise Tutoring 6 ANSWERS TO PRACTICE QUESTIONS CHANGING FROM TRANSFORMATIONAL FORM TO STANDARD FORM Quadratic equation in transformational form 2y 2 x 1 2 2y 3 x 12 2 STEP ONE STEP TWO Change the number in front of the left Change the number after the bracket to its reciprocal and move it to y value to its opposite and move it the front of the right bracket to the right hand side. y - y 3 1 x 122 y y 7 - 3 x 82 2 y - y 4 1 x 42 y 2 2 y 7 x 82 3 4y 4 x 4 2 © 2003 Paul Lobel Careerwise Tutoring 1 x 12 - 2 2 y 2 - 1 x 12 2 4 1 x 122 3 2 3 x 82 - 7 2 1 x 42 - 4 4 7 © 2003 Paul Lobel Careerwise Tutoring 8 EXAMPLE CHANGING FROM STANDARD FORM TO TRANSFORMATIONAL FORM Quadratic equation in standard form y - 3x 1 - 2 2 © 2003 Paul Lobel Careerwise Tutoring STEP ONE STEP TWO Change the number after the right Change the number in front of the bracket to its opposite and move it to right bracket to its reciprocal and the left side. move it in front of the left bracket. y 2 - 3x 1 2 1 y 2 x 12 3 9 PRACTICE QUESTIONS CHANGING FROM STANDARD FORM TO TRANSFORMATIONAL FORM Quadratic equation in standard form STEP ONE STEP TWO Change the number after the right Change the number in front of the bracket to its opposite and move it to right bracket to its reciprocal and the left side. move it in front of the left bracket. y - 7x 1 - 3 2 y 4x 9 6 2 y - 1 x 32 - 7 3 y - 0.5x 3 4 2 © 2003 Paul Lobel Careerwise Tutoring 2 10 ANSWERS TO PRACTICE QUESTIONS CHANGING FROM STANDARD FORM TO TRANSFORMATIONAL FORM Quadratic equation in standard form STEP ONE STEP TWO Change the number after the right Change the number in front of the bracket to its opposite and move it to right bracket to its reciprocal and the left side. move it in front of the left bracket. y - 7x 1 - 3 y 3 y 4x 9 6 y 6 4x 9 2 2 1 2 y - x 3 - 7 3 y - 0.5x 3 4 2 © 2003 Paul Lobel Careerwise Tutoring - 7x 1 2 2 y 7 y 4 - - 1 x 32 3 1 x 32 2 1 y 3 x 12 7 1 y 6 x 92 4 3y 7 x 3 2 2y 4 x 3 2 11 © 2003 Paul Lobel Careerwise Tutoring 12 EXAMPLE CHANGING FROM STANDARD FORM TO GENERAL FORM STEP ONE STEP TWO STEP THREE Simplify Quadratic equation in standard form FOIL the binomial square to Multiply number in front of get a perfect square trinomial bracket by numbers in bracket y - 3x 1 - 6 y -3x 2 2x 1 6 2 © 2003 Paul Lobel Careerwise Tutoring y -3x 2 6x - 3 6 y -3x 2 6x-9 13 PRACTICE QUESTIONS CHANGING FROM STANDARD FORM TO GENERAL FORM Quadratic equation in standard form STEP ONE STEP TWO STEP THREE FOIL the binomial square to Multiply number in front Simplify get a perfect square of bracket by numbers in trinomial bracket y 2x 1 - 4 2 y - 7x 1 - 5 2 y 1 x 62 3 2 y 1 x 82 -2 4 © 2003 Paul Lobel Careerwise Tutoring 3 14 ANSWERS TO PRACTICE QUESTIONS CHANGING FROM STANDARD FORM TO GENERAL FORM Quadratic equation in standard form y 2x 1 - 4 2 STEP ONE STEP TWO STEP THREE FOIL the binomial square to Multiply number in front Simplify get a perfect square of bracket by numbers in trinomial bracket y 2x 2 2x 1 4 y 2x 4x 2 4 2 y - 7x 1 - 5 y 7 x 2x 1 5 y -7x 14x - 7 5 2 2 1 2 1 2 x 12x 36 3 y y x 6 3 2 2 1 2 y x 8 - 2 4 © 2003 Paul Lobel Careerwise Tutoring y 1 2 x 16x 64 2 4 2 1 y x 2 6x 18 3 2 1 y x 2 4x 16 - 2 4 y 2x 2 4x - 2 y -7x 2 14x - 12 1 y x 2 6x 21 2 1 y x 2 4x 14 4 15 © 2003 Paul Lobel Careerwise Tutoring 16 EXAMPLE CHANGING FROM GENERAL FORM TO TRANSFORMATIONAL FORM ( when a = 1 ) Quadratic equation in general form y x 2 4x 7 STEP ONE STEP TWO STEP THREE Move constant "c" to opposite side and change sign. Add two blank spaces Find the missing number to make a trinomial perfect square and add this number to both sides. Simplify the left side and y 7 ___ x 2 4x ___ y 7 4 x 4x 4 2 factor the right side. y 11 (x 2)2 a=1 © 2003 Paul Lobel Careerwise Tutoring 17 PRACTICE PROBLEMS CHANGING FROM GENERAL FORM TO TRANSFORMATIONAL FORM ( when a = 1 ) Quadratic equation in general form STEP ONE STEP TWO STEP THREE Move constant "c" to opposite side and change sign. Add two blank spaces Find the missing number to complete the square and add it to both sides. Simplify the left side and factor the right side. y x 2 8x 4 y x 2 3x 18 y x 2 10x y x2 x 7 © 2003 Paul Lobel Careerwise Tutoring 18 ANSWERS TO PRACTICE PROBLEMS CHANGING FROM GENERAL FORM TO TRANSFORMATIONAL FORM ( when a = 1 ) STEP ONE STEP TWO STEP THREE Move constant "c" to opposite side and change sign. Add two blank spaces Find the missing number to complete the square and add it to both sides. Simplify the left side and y x 2 8x 4 y 4 ___ x 2 8x ___ y 4 16 x 2 8x 16 y x 3x 18 y 18 ___ x 2 3x ___ Quadratic equation in general form 2 9 9 y 18 x 2 3x 4 4 y x 2 10x y ___ x 2 10x ___ y 25 x 2 10x 25 y x2 x 7 y 7 ___ x 2 x ___ 1 1 y 7 x2 x 4 4 © 2003 Paul Lobel Careerwise Tutoring factor the right side. y 20 (x 4)2 81 3 y x 4 2 y 25 2 (x 5)2 27 1 2 y (x ) 4 2 19 © 2003 Paul Lobel Careerwise Tutoring 20 EXAMPLE CHANGING FROM GENERAL FORM TO TRANSFORMATIONAL FORM ( when "a" greater than 1 ) Quadratic equation in general form Move constant to left hand side and change its sign (+7) Factor the x terms so that x2 has a coefficient of 1. (Divide both x terms by 2 and put the 2 in front of brackets) Add blanks spaces to both sides as shown Find the missing number to make a perfect square trinomial in the right bracket. (4) Multiply the missing number by the number in front of the right bracket and add it to the left side. (Add 2 x 4 = 8) Simplify the left side by adding and change the trinomial to the square of a binomial (x + 2)2 Enclose the left had side in a bracket. Change the number in front of the right bracket to its reciprocal and put it in front the Careerwise left bracket. © 2003 Paul of Lobel Tutoring (1/2) y 2x 2 4x 7 y 7 2x 2 4x y 7 2 x 2 2x y 7 ___ 2x 2 2x ___ y 7 ___ 2x 2 2x 1 y 7 2 2x 2 2x 1 y 9 2x 1 2 1 y 9 x 12 2 21 PRACTICE PROBLEMS CHANGING FROM GENERAL FORM TO TRANSFORMATIONAL FORM ( when "a" greater than 1 ) Quadratic equation in general form y - 3x 2 6x 6 Move constant to left hand side and change its sign Factor the x terms so that x2 has a coefficient of 1. Add blanks spaces to both sides as shown Find the missing number to make a perfect square trinomial in the right bracket. Multiply the missing number by the number in front of the bracket and add it to the left side. Simplify the left side by adding and factor the trinomial to the square of a binomial Enclose the left had side in a bracket. Change the number in front of the right bracket to its reciprocal and put it in front the Careerwise left bracket. © 2003 Paul of Lobel Tutoring 5 22 ANSWER TO PRACTICE PROBLEM CHANGING FROM GENERAL FORM TO TRANSFORMATIONAL FORM ( when "a" greater than 1 ) Quadratic equation in general form y - 3x 2 6x 6 Move constant to left hand side and change its sign y 6 - 3x 2 6x Factor the x terms so that x2 has a coefficient of 1. y 6 - 3 x 2 2x Add blanks spaces to both sides as shown Find the missing number to make a perfect square trinomial in the right bracket. Multiply the missing number by the number in front of the right brackets and add the result to the left side. Simplify the left side by adding and factor the trinomial to the square of a binomial Enclose the left had side in brackets. Change the number in front of the right brackets to its reciprocal and put it in front of the left brackets. © 2003 Paul Lobel Careerwise Tutoring y 6 ___ - 3x 2 2x ___ y 6 ___ - 3x 2 2x 1 y 6 3 - 3x 2 2x 1 y 3 - 3x 1 2 1 y 3 x 12 3 23 PRACTICE PROBLEM CHANGING FROM GENERAL FORM TO TRANSFORMATIONAL FORM ( when "a" is not equal to 1 ) Quadratic equation in general form y 4x 2 2x 6 Move constant to left hand side and change its sign Factor the x terms so that x2 has a coefficient of 1. Add blanks spaces to both sides as shown Find the missing number to make a perfect square trinomial in the right bracket. Multiply the missing number by the number in front of the right brackets and add it to the left side. Simplify the left side by adding and factor the trinomial to the square of a binomial Enclose the left had side in a bracket. Change the number in front of the right bracket to its reciprocal and put it in front the Careerwise left bracket. © 2003 Paul of Lobel Tutoring 6 24 ANSWER TO PRACTICE PROBLEM CHANGING FROM GENERAL FORM TO TRANSFORMATIONAL FORM ( when "a" is greater than 1 ) Quadratic equation in general form y 4x 2 2x 6 Move constant to left hand side and change its sign y 6 4 x 2 2x Factor the x terms so that x2 has a coefficient of 1. 1 y 6 4 x 2 x 2 Add blanks spaces to both sides as shown Find the missing number to make a perfect square trinomial in the right bracket. Multiply the missing number by the number in front of the right brackets and add it to the left side. Simplify the left side by adding and factor the trinomial to the square of a binomial Enclose the left had side in a bracket. Change the number in front of the right bracket to its reciprocal and put it in front of the Careerwise left bracket. © 2003 Paul Lobel Tutoring 1 y 6 ___ 4 x 2 x ___ 2 1 1 y 6 ___ 4 x 2 x 2 16 1 1 1 y 6 4 x 2 x 4 2 16 y 23 1 4 x 4 4 2 1 23 1 y x 4 4 4 2 25