# Discriminant - Quadratic Formula

## Discriminant of the Quadratic Formula - How it Works - Video

### Definition

Definition of Quadratic Functions:

Here we have the standard form or general form of a quadratic function, f(x) = ax2 + bx + c, and a cannot be 0. If a is o, then we have f(x) = bx + c, which is a linear function.

### Discriminant Table

Discriminant Table:

The discriminant of the quadratic formula is the number under the radical sign, so b2 - 4ac. The discriminant could be positive, zero, or negative.

If the discriminant is positive, then there are 2 real solutions or 2 x-intercepts. If the discriminant is zero, then there is 1 real solution or 1 x-intercept. If the discriminant is negative, then there are no real solutions (2 imaginary solutions) or 0 x-intercepts.

### Example 1

Example 1:

Here we have our parabolic equation, 3x2 - 4x - 5 = 0.

Our first step is to solve for zero. Why zero? Well, we are solving for the zeros of the function, and another name for the zeros of the function are the x-intercepts or the roots of a function.

Now, we can label the a, b, and c values of our equation, so a is 3, b is -4, and c is -5. Now, we can put these numbers into the discriminant of the quadratic formula.

[ -b ± √ {b2 - 4ac} ] / (2a)

b2 - 4ac

(-4)2 - 4(3)(-5)

16 + 60

76

Quadratic Formula

Discriminant

Substituted the a, b, and c values

Multiplied

Combined like terms

Since our discriminant is positive, the graph of f(x) = 3x2 - 4x - 5 will cross the x-axis twice. So, we will have 2 real solutions.

### Example 2

Example 2:

Here we have our parabolic equation, -2x2 + 5x = 6.

Our first step is to solve for zero. Why zero? Well, we are solving for the zeros of the function, and another name for the zeros of the function are the x-intercepts or the roots of a function.

This time we have one step to do before we find a, b, and c. Right now we don't have 0 on one side. So we have to subtract 6 on both sides to get -2x2 + 5x - 6 = 0.

Now, we can label the a, b, and c values of our equation, so a is -2, b is 5, and c is -6. Now, we can put these numbers into the discriminant of the quadratic formula.

[ -b ± √ {b2 - 4ac} ] / (2a)

b2 - 4ac

(5)2 - 4(-2)(-6)

25 - 48

-23

Quadratic Formula

Discriminant

Substituted the a, b, and c values

Multiplied

Combined like terms

Since our discriminant is negative, the graph of f(x) = -2x2 + 6x - 6 will never cross the x-axis. So, we will have 0 real solutions.

### Example 3

Example 3:

Here we have our parabolic equation, 4x2 - 12x = -9.

Our first step is to solve for zero. Why zero? Well, we are solving for the zeros of the function, and another name for the zeros of the function are the x-intercepts or the roots of a function.

This time we have one step to do before we find a, b, and c. Right now we don't have 0 on one side. So we have to add 9 on both sides to get 4x2 - 12x + 9 = 0.

Now, we can label the a, b, and c values of our equation, so a is 4, b is -12, and c is 9. Now, we can put these numbers into the discriminant of the quadratic formula.

[ -b ± √ {b2 - 4ac} ] / (2a)

b2 - 4ac

(-12)2 - 4(4)(9)

144 - 144

0

Quadratic Formula

Discriminant

Substituted the a, b, and c values

Multiplied

Combined like terms

Since our discriminant is zero, the graph of f(x) = 4x2 - 12x + 9 will intersect the x-axis at one point. So, we will have 1 real solution.