Use the discriminant to determine the number of solutions of each quadratic equation.

Question

6x^2-2x+4=0

I need a benchmark answer and possibly a demonstration on how an answer would be achieved

Answer 1

For a quadratic equation written in the form ax²+bx+c=0, the quadratic formula is:

x= (-b±√(b²-4ac)/2a

The expression under the radical (b²-4ac) is called the discriminant.

If you evaluate the discriminant it's sign provides an indication as to the number and type of roots. Let's use the letter D to represent the value of the discriminant:

If D = 0, there is one rational root (of multiplicity 2).

If D > 0, there are two real solutions. If D is a perfect square the solutions are rational, otherwise irrational.

If D < 0, there are zero real solutions. However, when you start learning about imaginary and complex numbers there will be two complex solutions.

For your example:

a=6, b = -2, c = 4.

So, D = (-2)² - 4(6)(4) = 4-(4)(6)(4) = 4 - 96 = -92.

Since D < 0 there are no real solutions. That means if you graph the parabola y = 6x²-2x+4, you will see that it does not cross the x-axis:

desmos.com/calculator/mdt5hrn3f9

Answer 2

if the discriminant is

>0 then 2 real solutions

if

=0 then 1 real solution

if

<0 then 2 imaginary solutions, 0 real solutions