Find the values of k for which the following equations have real roots
$4x^2 + kx + 3 = 0$

Given:

Given quadratic equation is $4x^2 + kx + 3 = 0$.

To do:

We have to find the values of k for which the roots are real.

Solution:

Comparing the given quadratic equation with the standard form of the quadratic equation $ax^2+bx+c=0$, we get,

$a=4, b=k$ and $c=3$.

The discriminant of the standard form of the quadratic equation $ax^2+bx+c=0$ is $D=b^2-4ac$.

$D=(k)^2-4(4)(3)$

$D=k^2-48$

The given quadratic equation has real roots if $D≥0$.

Therefore,

$k^2-48≥0$

$k^2-(16\times3)≥0$

$k^2-(4\sqrt3)^2≥0$

$(k+4\sqrt3)(k-4\sqrt3)≥0$

$k≤-4\sqrt3$ and $k≥4\sqrt3$

The value of k can be represented as $(-∞, -4\sqrt3] U [4\sqrt3, ∞)$.

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Updated on: 10-Oct-2022

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