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

The value of k for which x2−4x+k=0{x^2} - 4x + k = 0 has coincident roots, is.
A) 4
B) -4
C) 0
D)-2

Explanation

Solution

Here, we have given a quadratic equationx2−4x+k=0{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=b2−4ac=0 = {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 x2−4x+k=0{x^2} - 4x + k = 0
Compare given quadratic equation with ax2+bx+c=0a{x^2} + bx + c = 0
So, we get a=1,a = 1, b = -4,c=k ,c = k
Hence, to find the value of k we use discriminant=b2−4ac=0 = {b^2} - 4ac = 0
Put the values of a, b, c.
=(−4)2−4(1)(k)=0= {\left( { - 4} \right)^2} - 4\left( 1 \right)\left( k \right) = 0
Simplify it.
⇒16−4k=0\Rightarrow 16 - 4k = 0
To find k, subtract 16 from the both sides.
⇒16−16−4k=0−16\Rightarrow 16 - 16 - 4k = 0 - 16
We get, ⇒−4k=−16 \Rightarrow - 4k = - 16
Divide both sides by (-4)
We get,⇒k=4 \Rightarrow k = 4
Thus, the value of k=4k = 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 : ax2 + bx + ca{x^{_2}}{\text{ }} + {\text{ }}bx{\text{ }} + {\text{ }}c. The discriminant, D = b2 − 4acD{\text{ }} = {\text{ }}{b^2}{\text{ }} - {\text{ }}4ac, have three cases:
a) If the discriminant is greater than zero i.e b2−4ac>0{b^2} - 4ac > 0 then the quadratic equation has two real, distinct (different) roots.
b) If the discriminant is less than zero i.e b2−4ac<0{b^2} - 4ac < 0 then the quadratic equation has no real roots.
c) If the discriminant is equal to zero i.e b2−4ac=0{b^2} - 4ac = 0 then the quadratic equation has two real, identical roots.