Question

Solve the following question given below:
If \[a{x^2} + bx + c = 0\], \[c \ne 0\] has equal roots, then what is the value of \[c\]?

Answer 1

Hint: This is the given quadratic equation. We find the discriminant of the given quadratic equation to know if the roots are real, equal or imaginary. Since it is given that the equation has real roots, we substitute the discriminant value for equal roots and find the value of the given variable.

Complete step-by-step solution:
In the given question, it is stated that the roots are equal.
Equal roots imply that the discriminant \[D\] should be equal to zero.
\[ \Rightarrow D = 0\]
The discriminant is given by the formula-
\[D = {b^2} - 4ac\]
Now substituting the discriminant value for real roots above, we get;
\[{b^2} - 4ac = 0\]
Now, we can arrange the above equation as given below;
\[4ac = {b^2}\]
To find the value asked, we get;
\[c = \dfrac{{{b^2}}}{{4a}}\]

Therefore the value of c is equal to \[\dfrac{{{b^2}}}{{4a}}\].

Additional Information: The discriminant of a quadratic equation is used to tell us if the roots of the equation are real, imaginary or equal. The nature of the roots is one case. The discriminant is also used to find if there are two solutions, one solution or no solutions to the given question. So, if we need to verify our answer, we simply can find the discriminant and check if the roots we acquired matches with the nature of roots and the number of solutions.

Note: Here, the discriminant should be considered zero because we were given that the roots are real. There should be a clear differentiation between real, imaginary and equal roots. We find the given variable by substituting the discriminant value in the given equation and writing in terms of the asked variable.