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?

Question

Kennedy Adams · Accepted Answer

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

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. 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]}

Hazel Anglin · Answer

undefined To find the discriminant of a quadratic equation $ ax^2 + bx + c = 0 $, the discriminant ($ \Delta $) is calculated using the formula:

$$ \Delta = b^2 - 4ac $$

In this case, the quadratic equation is $ 10x^2 - 2x + 2 = 3x - 4 $.

Comparing it with the standard form, we have:
$$ a = 10, \, b = -2, \, c = 2 - (3x - 4) = 6 - 3x $$

Substitute the values into the discriminant formula:
$$ \Delta = (-2)^2 - 4(10)(6 - 3x) $$
$$ \Delta = 4 - 240 + 120x $$
$$ \Delta = -236 + 120x $$

Now, to determine if the equation has one, two real, or two imaginary roots, we examine the value of the discriminant.

If $ \Delta > 0 $, the equation has two distinct real roots.
If $ \Delta = 0 $, the equation has one real root (a repeated root).
If $ \Delta < 0 $, the equation has two imaginary roots.

Therefore, for the equation $ 10x^2 - 2x + 2 = 3x - 4 $, the discriminant $ \Delta $ is $ -236 + 120x $. The nature of the roots depends on the value of $ x $ since $ \Delta $ is a function of $ x $.