Question

Find the value of \[p\] for which the quadratic equation \[4{x^2} + px + 3 = 0\] has equal roots.

Answer 1

Hint: Recall the formula to find the roots of a quadratic equation. Check the number of roots you would get from the equation. Try to find the value of \[p\] in such a way that the roots you obtained from the formula are equal.

Complete step-by-step answer:
Given, the equation as \[4{x^2} + px + 3 = 0\]
If \[a{x^2} + bx + c = 0\] is a quadratic equation, then roots of the equation is given by
 \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] (i)
Comparing the general equation with the given equation we get
 \[a = 4 \\
  b = p \\
  c = 3 \]
Now putting these values in equation (i), we get the roots of the equations as,
 \[x = \dfrac{{ - p \pm \sqrt {{p^2} - 4 \times 4 \times 3} }}{{2 \times 4}} \\
   \Rightarrow x = \dfrac{{ - p \pm \sqrt {{p^2} - 48} }}{8} \]
We get two roots,
 \[{x_1} = \dfrac{{ - p + \sqrt {{p^2} - 48} }}{8}\] and \[{x_2} = \dfrac{{ - p - \sqrt {{p^2} - 48} }}{8}\]
For the two roots to be equal, we observe that the term \[\sqrt {{p^2} - 48} \] of both the roots should be zero which would give us \[{x_1} = {x_2}\] .
Equating \[\sqrt {{p^2} - 48} \] to zero, we get
 \[\sqrt {{p^2} - 48} = 0 \\
   \Rightarrow {p^2} - 48 = 0 \\
   \Rightarrow {p^2} = 48 \]
 \[\Rightarrow p = \pm \sqrt {48} \\
   \Rightarrow p = \pm 4\sqrt 3 \]
Therefore, the value of \[p\] for which the roots are equal is \[ \pm 4\sqrt 3 \]
So, the correct answer is “ \[ \pm 4\sqrt 3 \] ”.

Note: We can know the number of roots of an equation from its type, like here it was a quadratic equation so there will be two roots. If it was a linear equation then there would be just one root.
In most of the questions you might have to find the roots of an equation, so always remember the formula to find the roots of a quadratic equation. The part \[\sqrt {{b^2} - 4ac} \] is called the discriminant of the quadratic equation, it tells us about the nature of the roots. If discriminant is less than zero then roots are unequal and imaginary, if discriminant is equal to zero then roots are equal and if it is greater than zero then roots are real and unequal.