Question

the value of k for which the roots of quadratic equations x²+4x+k=0 are real is​

Answer 1

Answer:

Hence, k must be less than 4 for the root of the equation \[{{x}^{2}}+4x+k=0\] to be real. Hence, \[k\le 4\] is the answer.

Answer 2

Answer:

Let us take ax2+ bx+c=0

to be a quadratic equation.

We know that the roots of the above equation are given by:

x=−b±b2−4ac−−−−−−−√2a

where, b2−4ac−−−−−−−√

is known as discriminant(D).

Now, comparing given equation x2+4x+k=0

with general equation ax2+ bx+c=0

, we will get:

a = 1, b = 4, and c = k.

Now, we will use formula to find the discriminant of equation x2+4x+k=0

given by:

D = b2−4ac−−−−−−−√

--------(1)

Here, we have a = 1, b = 4, and c = k

Putting the value of a, b, c in equation (1), we get

⇒D=(4)2−4(1)(k)−−−−−−−−−−−√

⇒D=16−4k−−−−−−√

Since, the roots of the above given equation in question i.e. x2+4x+k=0

is real. Hence, D≥0

∴16−4k−−−−−−√≥0

Squaring both side of the above inequality, we will get:

⇒(16−4k)≥0

Taking 4

common from the Left-hand side of the above inequality, we will get:

⇒4(4−k)≥0

It can also be written as:

⇒(4−k)≥0

Take k

from Left-hand to Right-hand side, we will get: -

⇒4≥k

⇒k≤4

Hence, k must be less than 4 for the root of the equation x2+4x+k=0

to be real.

Hence, k≤4

is the answer.