Quadratic Formula

TheQuadratic Formula

Review: What does it mean to
solve a quadratic equation?

These locations can be called:
• Zeroes
• Solutions
• X-intercepts
• Roots

Find where the function crosses the x-axis or where it
equals zero.

Review: Methods we know to
solve a quadratic equation

▪ Square Roots

▪ Graphing

NEW:
The Quadratic
Formula

What is the quadratic formula?
Why use it?

It is a formula that can be used to solve any
quadratic equation even when factoring is
impossible!

What are all
these variables?

They’re the coefficients
of a quadratic equation
in standard form.

a = 2
b = 3
c = 6

Let’s try an example:

1

Put it in standard form so the equation is set
equal to 0.

Let’s try an example:

2

Find a, b, and c.

a = 1
b = 5
c = 6

Let’s try an example:

3

Substitute it into the quadratic formula!

a = 1
b = 5
c = 6

Let’s try an example:

3

Substitute it into the quadratic formula! Remember to clean it up.

The solutions are: -2 and -3

Let’s Verify by graphing!

The solutions are: -2 and -3

The function crosses the
x-axis at -3 and -2.

Multiple Choice

What are the zeros of

$$f\left(x\right)=x^2+7x-18$$  ?

• $$x=-2,\ -9$$  

• $$x=2,\ -9$$  

• $$x=-2,\ 9$$  

• $$x=2,\ 9$$  

Multiple Choice

Solve using the quadratic formula:

4x² + 4x + 1 = 0

• x = -½

• x = -2

• x = 0

• x = ½

Multiple Choice

Use the quadratic formula to find the solutions for

y = -x² - 5x + 12

• No Real Solution

Important Terminology:
The Discriminant

The discriminant is the part of the
quadratic formula under the
square root.

What information can the
discriminant tell us?

It indicates the number of real solutions that the
function has.

We know that the square root of a positive number is
another positive number.

Therefore, if the discriminant is positive, there are two
real solutions.

We know that the square root of a negative number
gives us no real solution.

Therefore, if the discriminant is negative, there are
no real solutions.

We know that the square root of zero is zero.

Therefore, if the discriminant is zero, there is one real
solution.

Let’s try some examples:

How many real solutions does the function have?

Function

Discriminant

+/-/0 ?

# of Real Solutions

Negative!

Zero!

a = 6
b = 4
c = 9

-200

Let’s try some examples:

How many real solutions does the function have?

Function

Discriminant

+/-/0 ?

# of Real Solutions

Positive!

Two!

a = -2
b = 11
c = 3

145