Question

Find the value of \[k\], \[2{x^2} + kx + 3 = 0\] so that they have equal roots.

Answer 1

Hint: Here we need to find the value of the variable used in the quadratic equation. The given quadratic equation has equal roots, so the discriminant of the given equation will be zero. Then we will use the formula of determinant and equate the value of determinant with zero. From there, we will get the equation including the variable. After solving the obtained equation, we will get the required value of the variable.

Formula Used:
We will use the formula \[D = {b^2} - 4ac = 0\], where, \[b\] is the coefficient of \[x\], \[c\] is the constant term of equation and \[a\] is the coefficient of \[{x^2}\] .

Complete step-by-step answer:
The given quadratic equation is \[2{x^2} + kx + 3 = 0\]
It is also given that roots of the given quadratic equation are equal. So the discriminant of the given quadratic equation is equal to zero.
The general equation of a quadratic solution is \[a{x^2} + bx + c = 0\].
Now comparing the given equation to the general equation, we get
\[\begin{array}{l}a = 2\\b = k\\c = 3\end{array}\]
Substituting \[a = 2\],\[b = k\] and \[c = 3\] from the given quadratic equation in the formula of discriminant \[D = {b^2} - 4ac = 0\], we get
\[{k^2} - 4 \times 2 \times 3 = 0\]
On multiplying the terms, we get
\[ \Rightarrow {k^2} - 24 = 0\]
On factoring the equation using the algebraic identity \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\] , we get
\[ \Rightarrow \left( {k - 2\sqrt 6 } \right)\left( {k + 2\sqrt 6 } \right) = 0\]
This is possible when either \[k = 2\sqrt 6 \] or \[k = - 2\sqrt 6 \] .
Thus, the possible values of \[k\] are \[2\sqrt 6 \] and \[ - 2\sqrt 6 \]

Note: Here, we need to remember that the roots of any quadratic equation are two, and similarly the roots of the cubic equation are three and so on. Generally, the number of possible roots of any equation is equal to the highest power of the equation. The values of the roots of any equation always satisfy the equation i.e. when we put values of the roots in their respective equation, we get the value as zero.