Presentation on theme: "Solving Quadratic Equations by the Quadratic Formula"— Presentation transcript:

Do Now: Solve for x by factoring first:

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.

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

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

Graph the following: Trace to the approximate roots:

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

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

Practice: Solve:

Practice: Solve:

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.

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.

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

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 =

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=

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

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

Do Now Solve for x: Solve for x:

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

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

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

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

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

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

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

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

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

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

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

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.

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

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.

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

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

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!

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

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

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!

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!

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!

Example #1- continued Solve using the Quadratic Formula

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 =

Example #2- continued Solve using 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

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

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.