1. Solving for the Roots of a Quadratic Equation 12 October 2010

2. What do you know about solving quadratic equations?

3. Vocabulary Factor Factor –2 numbers whose product is the desired number –Usually occur in pairs –Example: Factors of 4 are 1, 4 but also 2 Factors of 4 are 1, 4 but also 2 –Finding factors on your calculator –Your turn: Find all possible the factors of 144 and 256 Find all possible the factors of 144 and 256

4. Vocabulary, cont. Factoring Factoring –The process of rewriting a number or an expression at the product of its factors –Examples: 4 = (1)(4) 4 = (1)(4) 4 = (2)(2) 4 = (2)(2)

5. Vocabulary, cont. Standard Quadratic Form f(x) = ax 2 + bx + c = 0, a ≠ 0 Leading Coefficient Constant

6. Why factor? Factoring allows us to solve for the x- values of a quadratic equation. Factoring allows us to solve for the x- values of a quadratic equation. We can find either the roots of the equation or the value of x for a specific value of y. We can find either the roots of the equation or the value of x for a specific value of y. By solving for the roots, we can graph a quadratic equation more easily by hand. By solving for the roots, we can graph a quadratic equation more easily by hand.

7. Preparing to Factor Make sure your equation is in standard quadratic form! Make sure your equation is in standard quadratic form! Make a list of possible factors (The list can be a mental list or on your calculator.) Make a list of possible factors (The list can be a mental list or on your calculator.)

8. Method 1: Greatest Common Factor (GCF) When we use it: When we use it: –When all the terms share at least one common factor (The terms can share more than one common factor.) –Generally, there isn’t a constant term.

9. Method 1: Steps for GCF Step 1: List the common factors of each of the terms in the equation/expression. Step 1: List the common factors of each of the terms in the equation/expression. Step 2: Factor out the common factors. Step 2: Factor out the common factors. Step 3 (if solving for the roots): Set the equation/expression equal to 0. Step 3 (if solving for the roots): Set the equation/expression equal to 0. Step 4: Solve for x by separating into two different equations equal to 0. Step 4: Solve for x by separating into two different equations equal to 0. Step 4: Discard any mathematically impossible answers, ex. 4 = 0. Step 4: Discard any mathematically impossible answers, ex. 4 = 0.

10. Method 1: GCF Examples Solve for the roots of the following equations: Solve for the roots of the following equations: 1.y = -x 2 + 6x

11. Method 1: GCF Examples, cont. 1. y = 21x 5 + 49x 3

12. Method 2: Difference of Squares When we use it: When we use it: –When the leading coefficient term (ax 2 ) and the constant term (c) are both perfect squares. –AND there is no middle term (bx).

13. Method 2: Steps for Difference of Squares Step 1: Find the square root of the leading coefficient term (ax 2 ). Step 1: Find the square root of the leading coefficient term (ax 2 ). Step 2: Find the square root of the constant term (c). Step 2: Find the square root of the constant term (c). Step 3: Write two sets of parentheses. Write the square roots of each term inside the parentheses. Step 3: Write two sets of parentheses. Write the square roots of each term inside the parentheses. Step 4: In the 1 st set of parentheses, write + in between the two terms. In the 2 nd set of parentheses, write — in between the two terms. Step 4: In the 1 st set of parentheses, write + in between the two terms. In the 2 nd set of parentheses, write — in between the two terms. Step 5: Set both sets of parentheses equal to 0 and solve for x. Step 5: Set both sets of parentheses equal to 0 and solve for x.

14. Method 2: Examples of Difference of Squares Solve for the roots of the following equations: Solve for the roots of the following equations: 1. y = x 2 - 25

15. Method 2: Examples of Difference of Squares, cont. 1. y = 4x 2 - 81

16. Method 2: Examples of Difference of Squares, cont. 1. y = 9x 2 - 36

17. Method 3: Box Method When we use it: When we use it: –When we can’t use GCF or the difference of squares.

18. Method 3: Steps in the Box Method Step 1: List the factors of the leading coefficient and the constant. Step 1: List the factors of the leading coefficient and the constant. Step 2: Pick a set of factors for both the leading coefficient and the constant. Step 2: Pick a set of factors for both the leading coefficient and the constant. Step 3: Cross multiply the numbers. Check if it’s possible to combine the products to create the middle term (b). Step 3: Cross multiply the numbers. Check if it’s possible to combine the products to create the middle term (b). Step 4: If not, pick a different set of factors and try the process again. If yes, then write the equation as the product of those factors. Step 4: If not, pick a different set of factors and try the process again. If yes, then write the equation as the product of those factors. Step 5: Set the equation equal to 0 and solve for x. Step 5: Set the equation equal to 0 and solve for x.

19. Method 3: Examples of the Box Method Solve for the roots of the following equations: Solve for the roots of the following equations: 1. y = x 2 + 7x + 6

20. Method 3: Examples of the Box Method, cont. 1. y = 2x 2 – 5x + 3

21. Method 3: Examples of the Box Method y = 4x 2 – 24x + 11 y = 4x 2 – 24x + 11

22. Method 4: Quadratic Equation When we use it: When we use it: –When an equation or expression can’t be factored.

23. Method 4: Steps for the Quadratic Equation Step 1: Make sure that the equation is in standard quadratic form! Step 1: Make sure that the equation is in standard quadratic form! Step 2: List the values for a, b, and c. Step 2: List the values for a, b, and c. Step 3: Substitute the appropriate values for a, b, and c into the formula for the quadratic equation. Step 3: Substitute the appropriate values for a, b, and c into the formula for the quadratic equation. Step 4: Solve for x. Remember that you will have two solutions! Step 4: Solve for x. Remember that you will have two solutions!

24. Method 4: Examples using the Quadratic Eqn Solve for the roots of the following equations: Solve for the roots of the following equations: 1. x 2 -12x + 4 = 0

25. Method 4: Examples using the Quadratic Eqn, cont. 1. x 2 + 6x + 3 = 0

26. Method 4: Examples using the Quadratic Eqn, cont. 1. 2x 2 – 7x – 5 = 0