1. Solving Quadratic Equations by the Quadratic Formula

2. Do Now:
Solve for x by factoring first:

3. THE QUADRATIC FORMULA
When you solve using completing the square on the general formula you get:
This is the quadratic formula! Equation must be in standard form first-ax2+bx+c=0
Just identify a, b, and c then substitute into the formula.

4. Do Using the Formula:
List a, b, c first:

5. Do Using the Formula:
List a, b, c first:

6. Graph the following:
Trace to the approximate roots:

7. Do Using the Formula:
List a, b, c first:

8. Do Using the Formula:
List a, b, c first:

9. Practice:
Solve:

10. Practice:
Solve:

11. WHY USE THE QUADRATIC FORMULA?
The quadratic formula allows you to solve ANY quadratic equation, even if you cannot factor it.
An important piece of the quadratic formula is what’s under the radical:
b2 – 4ac
This piece is called the discriminant.

12. WHY IS THE DISCRIMINANT IMPORTANT?
The discriminant tells you the number and types of answers
(roots) you will get. The discriminant can be +, –, or 0
which actually tells you a lot! Since the discriminant is
under a radical, think about what it means if you have a
positive or negative number or 0 under the radical.

13. WHAT THE DISCRIMINANT TELLS YOU!
Value of the Discriminant
Nature of the Solutions
Negative
2 imaginary solutions
Zero
1 Real Solution
Positive – perfect square
2 Reals- Rational
Positive – non-perfect square
2 Reals- Irrational

14. Example a=2, b=7, c=-11 Discriminant = Discriminant =
Find the value of the discriminant and describe the nature of the roots (real,imaginary, rational, irrational) of each quadratic equation. Then solve the equation using the quadratic formula) 1.
a=2, b=7, c=-11
Discriminant =
Value of discriminant=137
Positive-NON perfect square
Nature of the Roots – 2 Reals - Irrational
Discriminant =

15. The discriminant-b2-4ac
Students will review for the quadratics test.
The discriminant-b2-4ac
X2 -4x + 4 = 0
X2 -2x + 10 = 0
X2 +6x + 8 = 0
2X2 -5x + 1 = 0
Disc =
Disc=

16. The discriminant X2 -4x + 4 = 0 X2 -2x + 10 = 0 X2 +6x + 8 = 0
What is the nature of these roots?

17. The discriminant X2 -4x + 4 = 0 X2 -2x + 10 = 0 X2 +6x + 8 = 0
0, 1 real root
Negative = (2 imaginary roots)
pos perfect square= (2 rational roots)
pos not perfect=2 irrational roots

18. Do Now
Solve for x:
Solve for x:

19. Now add and multiply the 2 roots:
Solve for x:
Solve for x:

20. Sum and product of the roots:
Sum of the roots:
Product of the roots:

21. Sum and product of the roots:
Sum of the roots
Product of the roots:
Example:
Sum = 6
Product = 8

22. using sum and product: Do now:
Hint: we know what the sum is:
Given:
X2 -2x + c = 0 and one root is 5 find the other root
Find c

23. using sum and product: Do now:
Hint: we know what the sum is:
Given:
X2 -2x + c = 0 and one root is 1/3 find the other root

24. using sum and product: Do now:
Hint: we know what the product is:
Given:
X2 -bx + 10 = 0 and one root is 2, find the other root

25. Use the discriminant
For what value of c does the equation have equal roots:

26. Use the discriminant
For what value of c does the equation have equal roots:

27. Use the discriminant
For what value of c does the equation have imaginary roots:

28. Use the discriminant
For what value of c does the equation have imaginary roots:
(less than zero)

30. Quadratic equation knowing sum and product
X2 – sum (x) +product = 0
Write the equation when the sum is - 6 and the product is -8
Opposite of the sum, same as the product

31. using sum and product: to write an equation
Given the roots:
2, -5
Write the equation:
Hint: we can find the sum and the product.

32. using sum and product: to write an equation
Given the roots:
2, -5
Write the equation:
Sum= -3
Product = -10

33. using sum and product: Do now:
Given the roots:
2 + 6i,2-6i
Write the equation:
Hint: we can find the sum and the product.

34. Quadratic equation knowing sum and product
X2 – sum (x) +product = 0
Write the equation when the sum is 4 and the product is 40.

35. Quadratic equation knowing sum and product
X2 – sum (x) +product = 0
Write the equation when the sum is 2/3 and the product is 4

36. Quadratic equation knowing sum and product
X2 – sum (x) +product = 0
Write the equation when the sum is 2/3 and the product is 4
Clear the fraction!

37. using sum and product: Do now:
Given the roots:
Write the equation:
Hint: we can find the sum and the product.

38. using sum and product: Do now:
Given the roots:
Write the equation:
the sum is 2 and the product is 1/4.

39. Quadratic equation knowing sum and product
X2 – sum (x) +product = 0
Write the equation when the sum is 3/5 and the product 2/3
Clear the fraction!

40. Quadratic equation knowing sum and product
X2 – sum (x) +product = 0
Write the equation when the sum is 3/5 and the product -2/3
Clear the fraction!

41. Quadratic equation knowing sum and product
X2 – sum (x) +product = 0
Write the equation when the sum is 3/5 and the product -2/3
Clear the fraction!

42. Example #1- continued
Solve using the Quadratic Formula

43. Example 2 a=2, b=7, c=-11 Discriminant = Discriminant =
Find the value of the discriminant and describe the nature of the roots (real,imaginary, rational, irrational) of each quadratic equation. Then solve the equation using the quadratic formula) 1.
a=2, b=7, c=-11
Discriminant =
Value of discriminant=137
Positive-NON perfect square
Nature of the Roots – 2 Reals - Irrational
Discriminant =

44. Example #2- continued
Solve using the Quadratic Formula

45. Solving Quadratic Equations by the Quadratic Formula
Try the following examples. Do your work on your paper and then check your answers. Find the sum and product of the roots:
Sum Product

46. Solving Quadratic Equations by the Quadratic Formula
Try the following examples. Do your work on your paper and then check your answers. Find the sum and product of the roots:
Sum Product

47. Solving Quadratic Equations by the Quadratic Formula
Try the following examples. Do your work on your paper and then check your answers. Find roots:
Roots:

48. Solving Quadratic Equations by the Quadratic Formula
Try the following examples. Do your work on your paper and then check your answers.

49. Do Now:
Review-Students will review for the quiz on quadratic equations
Find the nature of the roots for the following quadratic equations:
Hint: use the discriminant.

50. Solving Quadratic Equations by the Quadratic Formula
Try the following examples.
Roots:

51. Solving Quadratic Equations using complete the
Try the following example.
Solve using complete the square: