Question

For what value of \[k\] , the roots of the quadratic equation \[kx\left( {x - 2\sqrt 5 } \right) + 10 = 0\] are equal?

Answer 1

Hint: We are given a quadratic equation and we are asked to find the value of \[k\] when the roots of the equation are equal. To solve this question, first recall the general expression for a quadratic equation and the formula to find the roots of a quadratic equation. Using this formula, find the roots of the equation and equate them to get the value of \[k\] .

Complete step-by-step answer:
Given, a quadratic equation \[kx\left( {x - 2\sqrt 5 } \right) + 10 = 0\] and the roots of this equation are equal.
A quadratic equation is an equation of second degree.
To find the value of \[k\] , let us simplify the equation to form an equation of second degree.
  \[kx\left( {x - 2\sqrt 5 } \right) + 10 = 0\]
 \[ \Rightarrow k{x^2} - 2k\sqrt 5 x + 10 = 0\] (i)
The general expression for quadratic equation is given by
 \[a{x^2} + bx + c = 0\] (ii)
where \[a\] , \[b\] and \[c\] are constants.
And the roots of a quadratic equation is given by
 \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] (iii)
Now, comparing equation (i) and (ii), we find the values of \[a\] , \[b\] and \[c\] as,
 \[a = k\]
 \[b = - 2k\sqrt 5 \]
 \[c = 10\]
Now, putting the values of \[a\] , \[b\] and \[c\] in equation (iii),
 \[x = \dfrac{{ - \left( { - 2k\sqrt 5 } \right) \pm \sqrt {{{\left( { - 2k\sqrt 5 } \right)}^2} - 4\left( k \right)\left( {10} \right)} }}{{2k}}\]
 \[ \Rightarrow x = \dfrac{{2k\sqrt 5 \pm \sqrt {20{k^2} - 40k} }}{k}\]
The roots will be,
 \[{x_1} = \dfrac{{2k\sqrt 5 + \sqrt {20{k^2} - 40k} }}{k}\] and \[{x_2} = \dfrac{{2k\sqrt 5 - \sqrt {20{k^2} - 40k} }}{k}\] (iv)
It is given that the roots are equal that is, \[{x_1} = {x_2}\] . We observe from equation (iv) that for \[{x_1} = {x_2}\] the term \[\sqrt {20{k^2} - 40k} \] must vanish or should be equal to zero.
Therefore equating the term \[\sqrt {20{k^2} - 40k} \] to zero, we get
 \[\sqrt {20{k^2} - 40k} = 0\]
 \[ \Rightarrow 20{k^2} - 40k = 0\]
 \[ \Rightarrow {k^2} - 2k = 0\]
 \[ \Rightarrow k(k - 2) = 0\]
 \[ \Rightarrow k = 0\,\,or\,\left( {k - 2} \right) = 0\]
 \[ \Rightarrow k = 0\,\,or\,k = 2\]
We get two values of \[k\] that are \[k = 0\] and \[k = 2\] .
Now if we put \[k = 0\] in the equation \[kx\left( {x - 2\sqrt 5 } \right) + 10 = 0\] then this would give us,
 \[0 \times x\left( {0 - 2\sqrt 5 } \right) + 10 = 0\]
 \[ \Rightarrow 10 = 0\] which is not true.
Therefore, the only possible value for \[k\] is \[2\] .
Hence, the required answer is \[2\] .
So, the correct answer is “k=2”.

Note: As, we have discussed above the formula to find the roots of a quadratic equation \[a{x^2} + bx + c = 0\] is \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] . The term \[D = {b^2} - 4ac\] is known as discriminant of a quadratic equation, discriminant of a quadratic equation tells us about the nature of the roots. If \[D < 0\] then the roots are imaginary roots, if \[D = 0\] the roots are equal and if \[D > 0\] the roots are real and distinct.