Question

The value of k for which ${x^2} - 4x + k = 0$ has coincident roots, is.
A) 4
B) -4
C) 0
D)-2

Answer 1

Hint:Here, we have given a quadratic equation${x^2} - 4x + k = 0$, has coincident root.Coincident roots are roots that are equal to each other.The roots are equal means the discriminant$ = {b^2} - 4ac = 0$.This information given to us.We can put the values of a, b, c in it & simplify it then we will get the value of k.

Complete step-by-step answer:
The given quadratic equation ${x^2} - 4x + k = 0$
Compare given quadratic equation with $a{x^2} + bx + c = 0$
So, we get $a = 1, $b = -4$ ,c = k$
Hence, to find the value of k we use discriminant$ = {b^2} - 4ac = 0$
Put the values of a, b, c.
$ = {\left( { - 4} \right)^2} - 4\left( 1 \right)\left( k \right) = 0$
Simplify it.
$ \Rightarrow 16 - 4k = 0$
To find k, subtract 16 from the both sides.
$ \Rightarrow 16 - 16 - 4k = 0 - 16$
We get, $ \Rightarrow - 4k = - 16$
Divide both sides by (-4)
We get,$ \Rightarrow k = 4$
Thus, the value of $k = 4$

So, the correct answer is “Option A”.

Note:The discriminant is a value calculated from a quadratic equation. It uses it to 'discriminate' between the roots (or solutions) of a quadratic equation. A quadratic equation is one of the form : \[a{x^{_2}}{\text{ }} + {\text{ }}bx{\text{ }} + {\text{ }}c\]. The discriminant, \[D{\text{ }} = {\text{ }}{b^2}{\text{ }} - {\text{ }}4ac\], have three cases:
a) If the discriminant is greater than zero i.e ${b^2} - 4ac > 0$ then the quadratic equation has two real, distinct (different) roots.
b) If the discriminant is less than zero i.e ${b^2} - 4ac < 0$ then the quadratic equation has no real roots.
c) If the discriminant is equal to zero i.e ${b^2} - 4ac = 0$ then the quadratic equation has two real, identical roots.