# How do you find the discriminant and how many and what type of solutions does 3x^2- 2x = 5 have?

May 3, 2015

$y = 3 {x}^{2} - 2 x - 5 = 0.$
For this type of quadratic equations, you don't need to find the Discriminant.
Shortcut: When a - b + c = 0, one real roots is (-1) and the other is $\left(- \frac{c}{a} = \frac{5}{3}\right) .$
2 real roots: -1 and 5/3.

May 3, 2015

For a quadratic equation in the form
$a {x}^{2} + b x + c = 0$
the discriminant is $\Delta = {b}^{2} - 4 a c$

Convert the given equation into the "standard form"
$3 {x}^{2} - 2 x = 5$

$\rightarrow 3 {x}^{2} - 2 x - 5 = 0$

$\Delta = {\left(- 2\right)}^{2} - 4 \left(3\right) \left(- 5\right)$

$\Delta = 4 + 60 = 64 = {8}^{2}$

Since $\Delta > 0$
the given equation has $2$ Real solutions

The full form for roots of the quadratic
$\frac{- b \pm \sqrt{\Delta}}{2 a}$

would become
$\frac{2 \pm 8}{2 \left(6\right)}$
and both solutions would be Rational (but not Integers)