# What is the discriminant of 2x^2 = 4x - 7 and what does that mean?

May 30, 2018

In the equation $a {x}^{2} + b x + c = 0$, the discriminant is ${b}^{2} - 4 a c$

#### Explanation:

By completing the square it is possible to see that the solutions of the equation:

$a {x}^{2} + b x + c = 0$

are of the form:

${x}_{1}$=$\frac{- b + \sqrt{{b}^{2} - 4 a c}}{2 a}$ and

${x}_{2}$=$\frac{- b - \sqrt{{b}^{2} - 4 a c}}{2 a}$

So, to have solutions in the real numbers (as opposed to complex numbers), the square root sqrt(b^2-4ac must exist as a real number, and so we need ${b}^{2} - 4 a c \ge 0$.

In summary, to have real solutions, the discriminant ${b}^{2} - 4 a c$ of the equation must satisfy ${b}^{2} - 4 a c \ge 0$