Question

If the roots of $2{x^2} - 6x + k = 0$ are real and equal, find $k$.

Answer 1

Hint: Here given a quadratic equation, and given that the quadratic equation has real and equal roots. If the quadratic equation is in the form of $a{x^2} + bx + c = 0$, then we know that the roots of this quadratic equation is given by $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$, here the roots are equal only if the under root in the numerator becomes zero. That is given by ${b^2} - 4ac = 0$, this expression is called the discriminant, then the quadratic equation has equal roots which are $\dfrac{{ - b}}{{2a}},\dfrac{{ - b}}{{2a}}$.
Hence if the quadratic equation has real and equal roots then its discriminant is equal to zero, given by:
$ \Rightarrow {b^2} - 4ac = 0$

Complete step-by-step answer:
Given a quadratic equation, for which the coefficient of ${x^2}$is given, the coefficient of $x$ is given, but the constant term in the quadratic equation is not given. The constant is assigned a variable named $k$.
We have to find the variable $k$.
Given that the quadratic equation has real and equal roots.
If the discriminant of the quadratic expression is equal to zero, then the quadratic expression has equal real and equal roots.
The discriminant of the given quadratic expression $2{x^2} - 6x + k = 0$ is given by:
$ \Rightarrow {\left( { - 6} \right)^2} - 4\left( 2 \right)\left( k \right) = 0$
$ \Rightarrow 36 - 8k = 0$
From the above expression determining the value of $k$, as given below:
$ \Rightarrow k = \dfrac{{36}}{8}$
$\therefore k = \dfrac{9}{2}$
Hence the value of $k$ is $\dfrac{9}{2}$.

The value of k is $\dfrac{9}{2}$.

Note:
Note that here we are asked to find the value of $k$, if the roots of the given quadratic equation are real and equal, from which we detected that the discriminant of the quadratic equation must be zero. But usually there are 3 general cases of the discriminant. Let the quadratic equation be $a{x^2} + bx + c = 0$
If the discriminant is greater than zero, ${b^2} - 4ac > 0$, then the quadratic equation has real and distinct roots.
If the discriminant is equal to zero, ${b^2} - 4ac = 0$, then the quadratic equation has real and equal roots.
If the discriminant is less than zero, ${b^2} - 4ac < 0$, then the quadratic equation has no real roots but has complex conjugate roots.