Question

Find the value of \[k\] for the following equation that has equal roots.
\[{x^2} + 4kx + \left( {{k^2} - k + 2} \right) = 0\]

Answer 1

Hint: First we have to define what the terms we need to solve the problem are. The roots have conditions and corresponding equations for them. By knowing the condition we start to solve the question.It is given that the given root is equal. By applying the condition of equal root we have a way to find the value of\[k\].
Formula to be used:
The roots may be imaginary, real, equal or unequal.
For an equation\[a{x^2} + bx + c = 0\], if\[{b^2} - 4ac = 0\], the roots are real and equal.
For an equation\[a{x^2} + bx + c = 0\], if \[{b^2} - 4ac > 0\]the roots are real and unequal.
For an equation\[a{x^2} + bx + c = 0\], if \[{b^2} - 4ac < 0\]the roots are imaginary.
Then we can write it as \[\Delta = {b^2} - 4ac\], and find the value of \[\Delta \].

Complete step by step answer:
The given equation is \[{x^2} + 4kx + \left( {{k^2} - k + 2} \right) = 0\]
We know that in the given equation the roots are equal. So we are applying the equal root condition.
From the given,
\[a = 1\] , \[b = 4k\] , \[c = \left( {{k^2} - k + 2} \right)\]
For equal roots, we know
\[\Delta = 0\]
Whereas\[\Delta = {b^2} - 4ac\],
\[{b^2} - 4ac = 0\]
By substituting the corresponding values in the given equation, we get the value of \[k\].
Substituting the values of \[a,b,c\]in the formula,
\[4{k^2} - 4(1)\left( {{k^2} - k + 2} \right) = 0\]
Multiplying \[4\] with the terms,
\[4{k^2} - 4{k^2} + 4k - 8 = 0\]
Simplifying we get,
\[4k - 8 = 0\]
Equating we get ,
\[4k = 8\]
By dividing,
\[k = \dfrac{8}{4}\]
Finally we get,
\[k = 2\]
When \[k = 2\] the roots are real and equal.
Hence the value of \[k\] is \[2\] .

Note:
The above condition\[{b^2} - 4ac = 0\]is applicable for only quadratic equation. The equation with degree \[2\] .
The condition is not applied for linear, cubic, and quartic equations.
We can also use this condition to find whether it is equal and real or unequal and real or imaginary.
We use this formula when the condition of the root is given.