Finding the Zeroes using Other Methods

Yesterday we looked at finding zeroes for quadratics that can factor into simple trinomials

Review Example Find the roots of y = x² - 10x – 24 We will write this equation in intercept form and then solve for x Recall that intercept form is y = a(x – s)(x – t)

Other ways we can find the roots of a function Graphically

What if it looks like our quadratic expression cannot be factored easily? This does not necessarily mean there are no roots The roots may be decimal values

The Quadratic Formula

We use the quadratic formula when it appears that our quadratic equation is not factorable – at least into two whole numbers This formula has the capability to tell us where our function crosses the x-axis, and how many times, even if our answer is a decimal

Yes, you can remember this formula Pop goes the Weasel e=related Gilligan’s Island re=related This one I can’t explain ure=related

The Quadratic Formula

How does it work Equation:

How does it work Equation:

y = 5x² + 6x + 1

y = -4x² + 3x – 2

y = 0.5x² + 2x - 1

y = 3x² - 6x + 4

The Discriminant Do you see b² - 4ac in the formula above? It is called the Discriminant, because it can "discriminate" between the possible types of answer: Let’s look closer at this part of the formula because it helps us when graphing quadratic equations

The Discriminant The discriminant is the number in the square root of the quadratic formula.

The Discriminant The Discriminant can be negative, positive or zero The value of the discriminant is what tells us how many times our function crosses the x-axis, or how many roots there are This can save us a lot of time if we don’t need to find the exact values!!!

If the Discriminant is positive, there are 2 real answers. This means the function crosses the x-axis two times

If the discriminant = 0, this means there is one place where the function crosses the x-axis In what cases will we see this? The only point it crosses the axis will be at the vertex

The Discriminant If the Discriminant is negative, there are 2 complex answers. This is where imaginary numbers come in to play We will say that this means there are no real solutions and our quadratic does not cross the x-axis In what cases will we see this?

Solve using the Quadratic formula

Describe the roots Tell me the Discriminant and the type of roots

Describe the roots Tell me the Discriminant and the type of roots 0, One rational root

Describe the roots Tell me the Discriminant and the type of roots 0, One rational root

Describe the roots Tell me the Discriminant and the type of roots 0, One rational root -11, Two complex roots

Describe the roots Tell me the Discriminant and the type of roots 0, One rational root -11, Two complex roots

Describe the roots Tell me the Discriminant and the type of roots 0, One rational root -11, Two complex roots 80, Two irrational roots