Question

Given α and 3α are the roots of the quadratic equation x² + kx + 27 = 0, what is the value of k?

A) k = -6
B) k = -9
C) k = -18
D) k = -27

Answer 1

The value of k in the quadratic equation x² + kx + 27 = 0 is -12. None of the option is correct

Step-by-step explanation:

Given that α and 3α are the roots of the quadratic equation x² + kx + 27 = 0, we can use the fact that the sum of the roots of a quadratic equation is equal to the opposite coefficient of x divided by the coefficient of x². So, α + 3α = -k/1 and 4α = -k. Since 4α = -k, we can substitute α with -k/4 in the equation x² + kx + 27 = 0. This gives us a new equation (-k/4)² + k(-k/4) + 27 = 0. Simplifying this equation, we get k²/16 - k²/4 + 27 = 0. Combining like terms and multiplying everything by 16, we have k² - 4k² + 432 = 0. This simplifies to -3k² + 432 = 0.

Finally, dividing both sides by -3, we find k² = -144. Taking the square root of both sides, we get k = ±12. Since we need the value of k, the correct answer is k = -12.

None of the option is correct