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?

1 Answer

Answer:

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

Explanation:

First, you must take the given equation and move it around to become #ax^2+bx+c =0# format.

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

#10x^2-5x+2=-4#

Now you add 4 to both sides.

#10x^2-5x+6=0#

Now, the formula to find the discriminant is: #Delta = b^2-4ac#
Using this equation, plug in what you have.

a = 10 b = -5 c = 6

Thus, you should have: #Delta =(-5)^2-4(10)(6)#

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]}