1. Factoring & Solving Quadratic Equations

2. Objectives : Factor quadratic expressions. Solve quadratic equations by factoring.

3. Standard form of the quadratic expression:

4. Examples of quadratic equations:

5. Factoring quadratics with a =1 : To factor.. Find two numbers that will multiply to equal the constant “ c ”, and at the same time add up to equal “ b” the coefficient of x-term x 2 + bx + c = ( x + m ) ( x + p ) m + p = b & mp = c

6. Factor: cb

7. CHECK!! (x+2)(x+3) = x 2 + 3x + 2x + 6 = x 2 + 5x + 6

8. Factor: Clues.. If c is (+) then the factors are either both (+) or both (-). If c is (-) then the factors are of alternating signs.  If b is (+) then they should be both (+).  If b is (-) then both should be (-).  If b is (+) then then the larger factor is (+).  If b is (-) then the larger is (-).

9. Factor: Now you try … 1) x 2 -9x +18 2) x 2 + 7x – 8 Click here to see answers

10. Factor: 1) (x – 6) ( x - 3) 2) (x + 8) ( x - 1)

11. Solving quadratic equations by factoring : Zero product property: Let a & b be real numbers or algebraic expressions. If : ab = 0 then a = 0 or b = 0

12. Solving quadratic equations by factoring : Follow the steps to solve quadratic equation by factoring: Write the equation in standard form. Factor completely. Use the zero-factor property and solve resulting equations. Check each solution in the original equation.

13. Solving quadratic equations by factoring :

20. Quadratic equations & Geometry ! Photography : rectangular photograph is 8 centimeters wide and 12 centimeters long. The photograph is enlarged by increasing the length and width by an equal amount in order to double its area. What are the dimensions of the new photograph?

21. Quadratic equations & Geometry ! Area of Rectangle = Length * Width 12 * 8 = 96 cm 2 Area after the enlargement ( doubled) 96 * 2 = 192 cm 2 Equation with length & width after increasing both dimensions by equal amount x: ( x + 12) * ( x + 8) = 192 8 cm 12 c m 8 + x 12 + x

22. Quadratic equations & Geometry ! simplify & write in quadratic form x 2 + 8x + 12 x + 96 -192 = 0 x 2 + 20x – 96 = 0 Factor (x + 24 ) ( x – 4) = 0

23. Quadratic equations & Geometry ! Zero Product property (x + 24 ) = 0 or ( x – 4) = 0 x = -24 or x = 4 But since dimensions are positive then -24 is ignored, and take only x = 4. Accordingly : Length = 12+(4)= 16 cm Width = 8 + (4) = 12 cm

24. Quadratic equation with k coefficient ! Determine all values of k, k < 0 so that the polynomial x 2 + kx + 12 can be factored using integers. Solution: To solve You need to factor.since k is an integers then look at the integer factors of 12. Factors of 12 : 1, 12, 2, 6 and 3, 4.

25. Quadratic equation with k coefficient ! But 12 is positive & k is negative, so we need subtraction signs in both factors. Possibilities.. x 2 + kx + 12 = ( x – 1) ( x – 12 ) x 2 + kx + 12 = ( x – 2) ( x – 6 ) x 2 + kx + 12 = ( x – 4 ) ( x – 4 )

26. Quadratic equation with k coefficient ! ( x – 1) ( x – 12 ) = x 2 – 13x + 12 ( x – 2) ( x – 6 ) = x 2 – 8x + 12 ( x – 4 ) ( x – 4 ) = x 2 - 7x + 12 So, the correct answers is the first one !! ( x – 1) ( x – 12 ) = x 2 – 13x + 12 so k = -13

27. Quadratics & real life ! Movie Theater : A company plans to build a large multiples theater.The financial analyst told he manager that the profit function for their theater was : P(x) = -x 2 + 48x – 512, where x is the number of movie screens & p(x) is the profit earned in thousands of dollars. Determine the range of production of movie screens that will guarantee that the company will not lose money.

28. Quadratics & real life ! p(x) = -x 2 + 48x – 512 -x 2 + 48x – 512 = 0 Write as an equation x 2 - 48x + 512 = 0 Multiply both sides by – 1 ( x – 16 ) ( x- 32 ) = 0 Factor ( x – 16 ) = 0 or ( x- 32 ) = 0 x = 16 or x = 32 So,the range is from 16 to 32 screens.