MATH 181-Quadratic Equations (7 ) 7.1 Solving a Quadratic Equation by Factoring I. Factoring Terms with Common Factors (Find the greatest common factor) a. 16 1x 4x = 4( 4 3x x ) 3 b. 14x y 35x y = 3 c. 1x y 4x y = II. Factoring by Grouping Pairs of terms have a common factor that can be removed in a process called factoring by grouping. *Hint: usually for 4 terms 3 a. x 3x 5x 15= x ( x 3) 5( x 3) =( x 3)( x 5) 3 b. p p 9p 18 = 3 c. y 3y 4y 1 = 4a a a d. 4x 1x 10x 30 = e. x ax bx ab = 3 f. x 5x x 10 g. 10m 1mn 5mn 30n 1
III. Factoring Trinomials Some trinomials can be factored into the product of two binomials. A. To factor a trinomial of the form x bx c, we look for two numbers with a product of c and a sum of b. a. z z 4 = ( z 6)( z 4) b. x x 0 = c. x 18x 81= d. y 4y 1= e. x y 18xy 64 = f. x 5xy 6y = g. 10 3x x = (hint: factoring -1 from the trinomial) h. a 8ab 33b = B. Factoring a trinomial of the form ax bx c, a 1 Use the trial factor method: use the factor of a and the factors of c to write all of the possible binomial factors of the trinomial. Then use FOIL to determine the correct factorization. To reduce the number of trial factors that must be considered, remember the following:
1. Use the sign of the constant term and the coefficient of x in the trinomial to determine the signs of the binomial factors. If the constant term is positive, signs of the binomial factors will be the same as the sign of the coefficient of x in the trinomial. If the sign of the constant term is negative, the constant terms in the binomials will have opposite signs.. If the terms of the trinomial do not have a common factor, then the terms in either one of the binomial factors will not have a common factor. a. Factor: 3x 8x 4, + factors of 3 - factors of 4 The terms have no common (coeff. of x ) (constant term) factor. 1, 3-1, -4 The constant term is positive. -, - Coefficient of x is negative. The binomial constant will be negative. 3
Write trial factors. Use the Trial Factors Middle Term Outer and Inner products of ( x 1)( 3x 4) 4x 3x 7x FOIL to determine the middle ( x 4)( 3x 1) x 1x 13x term of the trinomial. ( x )( 3x ) x 6x 8x Write the trinomial in factored form3x 8x 4 ( x )( 3x ) b. Trinomial of the ax bx x form can also be factored by grouping. To factor ax bx c, first find two factors a c whose sum is b. Then use factoring by grouping to write the factorization of the trinomial. 1. Factor: 3x 11x 8 Find two positive factors of 4 + Factors of 4 Sum (ac 3 8) whose sum is 11, the 1, 4 5 coefficient of x., 1 14 3, 8 11 The required sum has been found. The remaining factor need not be checked. Use the factors of 4 3x 11x 8 3x 3x 8x 8 whose sum is 11 to write ( 3x 3x) ( 8x 8) 4
11x as 3x 8x. = 3x( x 1) 8( x 1) Factor by grouping. ( x 1)( 3x 8) Check: ( x 1)( 3x 8) 3x 8x 3x 8 3x 11x 8. Factor: 4z 17z 1 Find two factors of -84 [ac 4 ( 1)] whose sum is - 17, the coefficient of z.. Factors of -84 Sum 1, - 84-83 - 1, 84 83, - 4-40 -, 4 40 3, - 8-5 - 3, 8 5 4, - 1-17 (Once the required sum is found, the remaining factors need not be checked). Use the factors of - 84 whose sum is -1 7 to write - 17z as 4z 1z. Factor by grouping. Recall that 1z 1 ( 1z 1) 4z 17z 1 4z 4z 1z 1 ( 4z 4z) ( 1z 1) z z z ( z 1)( 4z 1) 4 ( 1) 1( 1) 5
a. Factor: 6x 11x 10 b. Factor: 1x 3x 5 c. Factor: 30y xy 4x y d. Factor: 4x 15x 4 e. Factor: y 18y 7 f. Factor: 5 4x x g. Factor: 4a a 5 h. Factor: y 5 8y 3 15y i. Factor: 8x 8x 30 j. Factor: x 5x 1 6
IV. Special Factorizations 1. Factors of the Difference of Two Perfect Squares A B ( A B)( A B) a. x 5 ( x 5)( x 5) b. 9x 5 c. 4x 81y = d. 5x 1 = e. x 36y 4. Factors of a Perfect Square Trinomial A AB B ( A B) A AB B ( A B) a. x 8x 16 b. 9x 1x 4 c. 4x 0x 5 3. To Factor the sum or the difference of two cubes 3 3 A B ( A B)( A AB B ) 3 3 A B ( A B)( A AB B ) 7
a. a 64y 3 a ( 4y) 3 ( a 4y)( a 4ay 16y ) 4 b. 64y 15y c. 8x 3 y 3 z 3 3 3 d. x y 1 3 3 e. ( x y) x f. 7x 8y 3 6 g. 8y 64x 3 3 h. ( x ) 3 64 3 3 i. x y 3 3 j. x y k. x 3 7 3 3 l. b ( b 3) 3 9 m. y x n. 64 a 3 o. 5x 40y 3 3 8
V. Factor Polynomials using a Combination of Techniques a. 3x 7x 4 b. 3x y 4xy 48y c. 4x 6xy 16xy 4y d. 10a b 15ab 0b 4 e. x y 54xy f. 6x 3x 6y 9 g. 3x 18x 7 3y 5 h. 5x 45x i. ( a b) 18 1 j. x 3 7 k. a b 16ab 3 3 l. 3x 4x 48 3 m 3x 1x 36x n. 4y 36x 4 6 6 6 o. x y p. 6r s rs 1 4 4 q. x x y y 9
VI. Equation-Solving Principles A. If a b, then ac bc B. If a b, then a c b c C. If ab 0, then a 0 or b 0 D. If x k, then x k or x k E. If a b, then a n n b, for any positive integer n Solve for the unknown a. Solving by factoring: Example: 1. x 5x 6 0 ( x 6)( x 1) 0 so x 6 0 or x 1 0 so x = 6 or x = -1. x 9 0 Try to solve: a. x 6x b. 5 x 7 x 10
c. x 3 x 5 4 d. 10x 16x 6 0 e. 3 4m 4 m m m 4 f. y 5 y 3 1 g. Find the x-intercepts of y x x 8 h. Find two positive integers that have sum of 10 and product of 1. k. A particular school is planning on extending the building. If the current building is rectangular and measures 35 feet by 10 feet. The planned extension will double this area and is to be carried out uniformly on the north and est sides. How much should the building be extended? 11
7. Solving Quadratic Equation by Completing the Square I. Quadratic Equations Quadratic equation is an equation equivalent to ax bx c 0, a 0 such that the coefficients a, b, c are real numbers. II. Quadratic Functions Quadratic function f is a second-degree polynomial function f ( x) ax bx c are real numbers. III. Solving Quadratic Functions A. Solving by factoring: Example: 1. x 5x 6 0, a 0, such that a, b, and c ( x 6)( x 1) 0 so x 6 0 or x 1 0 so x = 6 or x = -1. x 9 0 1
S. Nunamaekr B. Solving by Completing the Square Example: Solve for the unknown 1. x 6x 10 0 x 6x 10 x 6 6 6x ( ) 10 ( ) x x 6x 9 10 9 6x 9 19 6 6 ( x )( x ) 19 ( x 3) 19 ( x 3) 19 x 19 3 or 3 19 To solve a quadratic equation by completing the square: 1. Isolate the terms with variables on one side of the equation and arrange them in descending order.. Divide by the coefficient of the squared term if that coefficient is not 1. 3. Complete the square by taking half the coefficient of the first-degree term and adding its square on both sides of the equation. 4. Express one side of the equation as the square of a binomial 13
5. Use the principle of square roots. 6. Solve for the variable. Example: Solve for the Unknown 1. Solve: x 1 3x. Solve: x 3x 10 0 14
7.3 Solving Quadratic Equations by the Quadratic Formula Consider any quadratic equation in standard form: ax bx c 0, a 0 x b b 4ac a *Discriminant is b 4ac if b if b if b 4ac 0 --> one real number solution 4ac 0 two real-number solutions 4ac 0 two imaginary-number solutions,complex conjugates Example: Solve: x 5x 8 0 x 5 5 4 ( 1 )( 8 ) 5 7 ( 1) i x 5 7 15
Example: Solve by quadratic formula 1. 3x 5x 0. y 3y 0 3. 5m 3m 4. 3x 4 5x 16
7.4 Applications of Quadratic Equations I. Guidelines for Applying Mathematics 1. Summarize the situation. Pinpoint the unknown ( often represented by x).. Draw a sketch (if possible) 3. Decide on a strategy, and find an equation in x. 4. Solve the equation for x. 5. Interpret the result. II. Examples 1. An object is propelled vertically from the ground level with velocity v. We have already seen that the height s at time t is given by the equation s 16t vt. If the initial velocity is 56 ft per second, when will the object be at a height of 768 ft?. Find two positive integers that have sum of 10 and product of 1. 3. A particular school is planning on extending the building. If the current building is rectangular and measures 35 feet by 10 feet. The planned extension will double this area and is to be carried out uniformly on the north and est sides. How much should the building be extended? 4. A rectangle is.00 ft longer than it is wide. Find its dimensions if the area is 48.0 ft 17