Question

Find the value of k for which 4x^3+kx-27 = 0 has two equal roots

Answer 1

4(x-a)(x-b)^2 = 4(x^3 + mx - 27/4) must have no x^2 term. (k=4m)

Also, ab^2 = 27/4 = 3^3 so
a = 3 and b^2 = 9/4
a quick check shows that b = -3/2 works.
(x-3)(2x+3)^2 = 4x^3 - 27 x - 27
so k = -27

Answer 2

[AI-generated answer — Explain Bot]

To find the value of k for which the equation 4x^3 + kx - 27 = 0 has two equal roots, we need to use the concept of discriminant of a quadratic equation.

Let's first rewrite the equation in the form of a quadratic equation: 4x^3 + kx - 27 = 0.

The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by the formula: D = b^2 - 4ac.

In this case, since we have a cubic equation, the discriminant will be slightly different. The equation 4x^3 + kx - 27 = 0 can be rewritten as 4x^3 + 0x^2 + kx - 27 = 0, with a = 4, b = 0, and c = k.

Let's calculate the discriminant of this cubic equation:
D = b^2 - 4ac = 0^2 - 4(4)(k) = -16k.

Now, for this equation to have two equal roots, the discriminant needs to be equal to zero (D = 0), since the two equal roots of a quadratic equation are obtained when the discriminant is zero.

Therefore, we have -16k = 0.

To solve this equation for k, divide both sides by -16:
k = 0 / -16
k = 0

So, the value of k for which the equation 4x^3 + kx - 27 = 0 has two equal roots is k = 0.