Question

How many real number solutions does the equation have?
y=5x^2+2x-12

Answer 1

Do you know about the use of the discriminant of a quadratic equation?

b^2 - 4ac

the letters come from the general form

ax^2 + bx + c = 0

Answer 2

A simple way to find how many solutions it has is by using the discriminant, the discriminant is b^2 - 4ac

A = 5
B = 2
C = -12

So, 2^2 -4(5)(-12) = 4 - 240 = -236

Now because it's under the radical, there is NO Real number that could give you a number times itself to be negative, what I mean is a square is a number times itself, so 4 * 4 = 16, works. -4 * -4 = 16, also works, but -4 * 4 = -16, it's |9\ \sf NOT\) times itself therefore it's considered to have no solutions.

Now, using complex numbers this would be i√236, but that's a different story.

If:

\(\ \sf \Large b^2 - 4ac = 0 \longrightarrow it ~ has ~ only ~ one ~ solution\)
\(\ \sf \Large b^2 - 4ac > 0 ~ it ~ will ~ have ~ 2 ~ solutions\) if it A,B and C are rational then your soltions will be rational.
\(\ \sf \Large b^2 - 4ac < 0\) such as negative numbers, \(\ \sf \Large It ~ will ~ have ~ NO ~ Solutions \)

Answer 3

4 * 4 = 16, works. -4 * -4 = 16, also works, but -4 * 4 = -16, it's \(\ \sf NOT\) times itself therefore it's considered to have no solutions. ;_; It didn't show up :\