Answer
$\text{Discriminant is }4$.
$\text{There are two real solutions to the equation.}$
Work Step by Step
In a quadratic equation, which takes the form $ax^2 + bx + c = 0$, the discriminant is found within the Quadratic Formula.
The Quadratic Formula is given as:
$x = \frac{-b ± \sqrt {b^2 - 4ac}}{2a}$, where $a$ is the coefficient of the squared term, $b$ is the coefficient of the linear term, and $c$ is the constant term.
The expression $b^2 - 4ac$ within the radical is called the discriminant. The discriminant can determine how many solutions there are to the quadratic equation and whether the roots are real or imaginary.
In this exercise, $a = 1$, $b = -12$, and $c = 30$. Plug these values into the discriminant:
$(-12)^2 - 4(1)(30)$
Evaluate the exponent first:
$144 - 4(1)(30)$
Do the multiplication:
$144 - 120$
Subtract:
$24$
If the discriminant is positive, the equation has two real solutions.
If the discriminant is zero, there is one real solution.
If the discriminant is negative, then there are two imaginary solutions.
In this exercise, the discriminant is positive; therefore, there are two real solutions to the equation.