Courses
Courses for Kids
Free study material
Free LIVE classes
More

# The quadratic equation $kx\left( {x - 2} \right) + 6 = 0$ has two equal roots. Find the value of k.

Last updated date: 20th Mar 2023
Total views: 304.8k
Views today: 4.84k
Verified
304.8k+ views
Hint: Here, we will be putting the discriminant of the given quadratic equation equal to zero in order for this equation to have two equal roots. From here we will obtain the value of unknown k.

The given quadratic equation is $kx\left( {x - 2} \right) + 6 = 0 \\ \Rightarrow k{x^2} - 2kx + 6 = 0{\text{ }} \to {\text{(1)}} \\$
As we know that for any general quadratic equation $a{x^2} + bx + c = 0{\text{ }} \to {\text{(2)}}$ to have two equal roots, the discriminant should be equal to zero where the discriminant is given by
${\text{d}} = \sqrt {{b^2} - 4ac} \to {\text{(3)}}$
$\Rightarrow \sqrt {{{\left( { - 2k} \right)}^2} - 4\left( k \right)\left( 6 \right)} = 0 \\ \Rightarrow \sqrt {4{k^2} - 24k} = 0 \\ \Rightarrow 4{k^2} - 24k = 0 \\ \Rightarrow 4k\left( {k - 6} \right) = 0 \\$
Either $4k = 0 \\ \Rightarrow k = 0 \\$ or $\left( {k - 6} \right) = 0 \\ \Rightarrow k = 6 \\$
But here k=0 is neglected because if k=0 the given quadratic equation $kx\left( {x - 2} \right) + 6 = 0$ reduces to 6=0 which is not true.
Note: In these types of problems, the values of the roots corresponding to the any quadratic equation $a{x^2} + bx + c = 0$ are given by $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$. If the quadratic equation has two equal roots then the value of that root is given by $x = \dfrac{{ - b}}{{2a}}$.