Question

In the equation x^2+kx+12 one root is twice the other root. What are the value(s) of k?

Answer 1

If the roots are m and 2m, then

(x-m)(x-2m) = x^2-3m+2m^2
so 2m^2 = 12, and m^2 = 6
(x-√6)(x-2√6) = x^2 - 3√6 x + 12
Looks like k = ±3√6

Answer 2

or

using x^ - (sum of roots)x + (product of roots) = 0 property
using ooblecks roots of m and 2m
sum of roots = m+2m = 3m
product of the roots = 2m^2
2m^2 = 12 or m = ±√6

k = -3m = ±3√6

Answer 3

[AI-generated answer — Step-by-Step Bot]

To find the value(s) of k in the equation x^2+kx+12, where one root is twice the other root, we can use the sum and product of roots formula.

Let's assume that one root is a and the other root is 2a.

According to the sum of roots formula, the sum of roots in a quadratic equation ax^2+bx+c = 0 is given by -b/a. In our equation, the sum of roots is a + 2a = 3a.

According to the product of roots formula, the product of roots in a quadratic equation ax^2+bx+c = 0 is given by c/a. In our equation, the product of roots is (a)(2a) = 2a^2.

Since the given equation is x^2+kx+12, we can equate the sum of roots and the product of roots to their corresponding values in the equation:

3a = -k (sum of roots formula)
2a^2 = 12 (product of roots formula)

Simplifying the second equation, we have:

2a^2 = 12
Divide both sides by 2:
a^2 = 6
Taking the square root of both sides, we get:
a = √6

Substituting this value of a in the equation 3a = -k, we have:

3√6 = -k

To find the value of k, we can multiply both sides by -1:

-k = -3√6

Now, multiplying both sides by -1, we get:

k = 3√6

Therefore, the value of k is 3√6.

Answer 4

[AI-generated answer — Explain Bot]

To find the value(s) of k in the equation x^2 + kx + 12, where one root is twice the other root, we can use the properties of quadratic equations.

Let's assume the two roots of the equation are r and 2r (since one root is twice the other). By the quadratic formula, the sum of the roots (r + 2r) is equal to -k, and the product of the roots (r * 2r) is equal to 12.

Sum of the roots:
r + 2r = -k
3r = -k

Product of the roots:
r * 2r = 12
2r^2 = 12

Now, we can solve these two equations simultaneously to find the value(s) of k.

From the first equation, we can express r in terms of k:
3r = -k
r = -k/3

Substituting this value of r into the second equation, we have:
2r^2 = 12
2(-k/3)^2 = 12
2(k^2/9) = 12
(k^2/9) = 6
k^2 = 6 * 9
k^2 = 54

Taking the square root of both sides:
k = ±√(54)

So, the value(s) of k are ±√(54), which simplifies to ±3√6.

Therefore, the possible values of k are ±3√6.