Determine the values of K such that the quadratic equation x^2 + 2Kx - 3K =0 has equal roots

Thankyou

why plug in 3 for x?

Why change the polynomial?

x^2 + 2kx - 3k = 0
the discriminant is (2k)^2 - 4(1)(-3k) = 4k^2 + 12k

For two equal roots, the discriminant must be zero.

4k^2+12k = 4k(k+3) = 0
k = 0 or -3

k=0: clearly x^2=0 has two equal roots
k = -3: x^2-6x+9 = (x-3)^2 has two equal roots.

2x^2 - kx - 3=0

just plug in 3 for x and solve for k
2*3^2 -3k - 3=0
2*9-3k-3=0
15=3k
5=k

I agree with you messed up my math.

To determine the values of K such that the quadratic equation x^2 + 2Kx - 3K = 0 has equal roots, we can use the discriminant of the quadratic equation.

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

If the discriminant is equal to zero (Δ = 0), then the quadratic equation will have equal roots.

So in our given equation, x^2 + 2Kx - 3K = 0, we can substitute a = 1, b = 2K, and c = -3K into the discriminant formula.

Δ = (2K)^2 - 4(1)(-3K) = 4K^2 + 12K = 4K(K + 3)

For the equation to have equal roots, Δ must be equal to zero. Therefore, we set 4K(K + 3) = 0 and solve for the values of K:

4K(K + 3) = 0

This equation can be satisfied in two ways:

1) 4K = 0 => K = 0
2) K + 3 = 0 => K = -3

Therefore, the values of K that make the quadratic equation have equal roots are K = 0 and K = -3.

So, when K is either 0 or -3, the quadratic equation x^2 + 2Kx - 3K = 0 will have equal roots.