Question

Find the value of k for which the quadratic equation \[\left( {k + 4} \right){x^2} + \left( {k + 1} \right)x + 1 = 0\] has equal roots.

Answer 1

Hint: It has given that the quadratic equation has equal roots. So this means the determinants will be zero. i.e. \[{b^2} - 4ac\]= 0, where a, b and c are the coefficients of \[{x^2}\], x and constant terms of the given quadratic equation in the question. And solve the quadratic equation of k to get the required value of k.



Complete step-by-step answer:
As we know that if the highest degree (highest power of the variable) of any polynomial equation is 2 then the polynomial equation will be quadratic.
And we know that the general quadratic equation is written as,
\[a{x^2} + bx + c = 0\] ----- (1)
And according to the formula for the roots of the quadratic.
Roots of general quadratic equations are equal to \[\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\].
And as we know that if the roots of this quadratic equation are equal then \[{b^2} - 4ac\] must be equal to zero.
Now the given quadratic equation is,
\[\left( {k + 4} \right){x^2} + \left( {k + 1} \right)x + 1 = 0\] -----(2)
So, now we had to find the value of a, b and c for the given quadratic equation.
After that we can apply the direct formula and find the value of \[{b^2} - 4ac\] for the given quadratic equation and equate that with zero.
So, comparing equation 2 with equation 1. We get,
\[a = k + 4,{\text{ }}b = k + 1\] and \[c = 1\]
Now it is given that the roots of the given quadratic equation are equal. So, \[{b^2} - 4ac\] = 0
Now finding the value of \[{b^2} - 4ac\]. We get,
\[{b^2} - 4ac\] = \[{\left( {k + 1} \right)^2} - 4\left( {k + 4} \right)1\] = 0
\[\left( {{k^2} + 2k + 1} \right) - \left( {4k + 16} \right)\] = 0 -----(3)
Now we had to solve equation 3 to get the value of k.
\[{k^2} - 2k - 15 = 0\]
Now let us split the middle term of the above equation and then solve it. We get,
\[
  {k^2} - 5k + 3k - 15 = 0 \\
  k\left( {k - 5} \right) + 3\left( {k - 5} \right) = 0 \\
 \]
Taking (k – 5) common from the above equation. We get,
\[\left( {k - 5} \right)\left( {k + 3} \right) = 0\]
So, k = 5 or k = – 3
Hence, for the value of k = 5 or – 3, the roots of the given quadratic equation will be equal.

Note:- Whenever we come up with this type of problem then first, we have to write the general quadratic equation and then by comparing that with the given equation we will find the value of a, b and c. And after that we find the equation in terms of k using the condition that if the roots of the quadratic equation are equal then \[{b^2} - 4ac\] must be equal to zero. And after solving this equation we will get the required value of k.