# How can you use the discriminant to find out the nature of the roots of 7x^2=19x ?

Oct 26, 2016

Check the sign and value of the discriminant to find that the given equation will have two rational solutions.

#### Explanation:

Given a quadratic equation $a {x}^{2} + b x + c = 0$, the discriminant is the expression ${b}^{2} - 4 a c$. Evaluating the discriminant and observing its sign can show how many real solutions the equation has.

• If ${b}^{2} - 4 a c > 0$, then there are $2$ solutions.
• If ${b}^{2} - 4 a c = 0$, then there is $1$ solution.
• If ${b}^{2} - 4 a c < 0$, then there are $0$ solutions.

The reasoning behind this is clear upon looking at the quadratic formula:

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

If the discriminant is positive, then two distinct answers will be obtained by adding or subtracting its square root. If it is $0$, then adding or subtracting will lead to the same single answer. If it is negative, it has no real square root.

Putting the given equation in standard form, we have

$7 {x}^{2} = 19 x$

$\implies 7 {x}^{2} - 19 x + 0 = 0$

$\implies {b}^{2} - 4 a c = {\left(- 19\right)}^{2} - 4 \left(7\right) \left(0\right) = {19}^{2} > 0$

Thus, there are two solutions.

We can also find whether the solution(s) will be rational or irrational. If the discriminant is a perfect square, then its square root will be an integer, making the solutions rational. Otherwise, the solutions will be irrational.

In the above case, ${b}^{2} - 4 a c = {19}^{2}$ is a perfect square, meaning the solutions will be rational.

(If we solve the given equations, we will find the solutions to be $x = 0$ or $x = \frac{19}{7}$)