Question

How to find solutions to an unknown variable using the discriminant?

Answer 1

Hint: The discriminant is defined for the solutions of a quadratic equation when expressed in the standard form. It is used for determining the nature of the roots of the quadratic equation. The nature of roots can be imaginary, real and equal, or real and distinct. After determining the nature of the roots, it is substituted into the quadratic formula to get the final solutions.

Complete step-by-step solution:
Since we are told about the discriminant in the above question, this means that the question is concerned with the quadratic equations, whose standard form is given by
$a{x^2} + bx + c = 0$
Now, we know that the solutions of the above equation are determined by the quadratic formula, which is given by
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Now, the quantity inside the square root is known as the discriminant, that is,
$D = {b^2} - 4ac$
Now, for finding the solutions of the unknown variable x of a quadratic equation, first we determine its nature by calculating the discriminant. On the basis of the values of the discriminant, three cases are possible:
(i) $D > 0$: In this case the two roots are real and distinct.
(ii) $D < 0$: In this case, the roots are imaginary, and occur in the conjugate pairs.
(iii) $D = 0$: In this case, the roots are equal.
After evaluating the discriminant, we substitute it into the quadratic formula written above to find the final solutions of the unknown variable.

Note:
A quadratic equation is an equation that has the highest degree of 2 and has 2 solutions. We can find the solutions or roots of the equation by using methods, such as the middle term splitting, factorization method, etc. But we must note that these work only when the discriminant is a perfect square. So these methods cannot be used to solve a quadratic equation for the general case. Only quadratic formulas can be used to find the solution of any type of quadratic equation.