# How do you find the discriminant of 10x^2-2x+2=3x-4 and use it to determine if the equation has one, two real or two imaginary roots?

Mar 7, 2017

Discriminant = $- 215$
Therefore, there are two imaginary solutions.

#### Explanation:

First, you must take the given equation and move it around to become $a {x}^{2} + b x + c = 0$ format.

To do this, you must start off my subtracting $3 x$ from both sides. You're left with:

$10 {x}^{2} - 5 x + 2 = - 4$

Now you add 4 to both sides.

$10 {x}^{2} - 5 x + 6 = 0$

Now, the formula to find the discriminant is: $\Delta = {b}^{2} - 4 a c$
Using this equation, plug in what you have.

a = 10 b = -5 c = 6

Thus, you should have: $\Delta = {\left(- 5\right)}^{2} - 4 \left(10\right) \left(6\right)$

Your answer equals: $\Delta = - 215$

Here is a key to find out the type of answer you'll receive.

• If the discriminant: $\Delta < 0$ you will have 2 imaginary solutions.
• If the discriminant: $\Delta = 0$, you will have one real answer.
• If the discriminant: $\Delta > 0 ,$ you will have two real solutions.

Because $- 215 < 0 ,$ you will have 2 imaginary solutions.

You can even check a graph and see that because the parabola never touches the x-axis, there are no real solutions, but two imaginary:
graph{10x^2-5x+6 [-16.05, 16.04, -8.03, 8.02]}