How can you use the discriminant to find out the nature of the roots of #7x^2=19x# ?
Check the sign and value of the discriminant to find that the given equation will have two rational solutions.
Given a quadratic equation
#b^2-4ac > 0#, then there are #2#solutions.
#b^2-4ac = 0#, then there is #1#solution.
#b^2-4ac < 0#, then there are #0#solutions.
The reasoning behind this is clear upon looking at the quadratic formula:
If the discriminant is positive, then two distinct answers will be obtained by adding or subtracting its square root. If it is
Putting the given equation in standard form, we have
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,
(If we solve the given equations, we will find the solutions to be