If the sum of the zeroes of the polynomial p(x)=(k^2 - 14) x^2 - 2x - 12 is 1, find the value of k?

Question

Answer

Answer: The values of k are -4 and 4

Step-by-step explanation:

The given polynomial is (k²-14)x²-2x-12

The sum of zeros of a quadratic equation is -b/a.

Hence, the sum of the zeros of the given quadratic is
-(-2)/(k²-14)
2/(k²-14)

Now, from the given directions, this expression should be equal to 1.
2/(k²-14) = 1
k²-14 = 2
k² = 16
k = √±4

Answer

P(x)= (x^2-14)x^2-2x-12,
sum of zeros of the polynomial=1---( given),
or, -b/a=1,
-(-2)/k^2-14=1,
k^2-14=2,
k^2=16,
k=+4,-4

Answer

Problem: Find the value of k if the sum of the zeroes of p(x) = (k^2 - 14)x^2 - 2x - 12 is 1.

Solution:

Step 1: Find the sum of the zeroes of the polynomial
Let the zeroes of the polynomial be a and b. Then, by Vieta's formulas, we know that:

a + b = -(-2)/(k^2 - 14) = 2/(k^2 - 14)

Given that the sum of the zeroes is 1, we can write:

a + b = 1

Substituting this into the previous equation, we get:

1 = 2/(k^2 - 14)

Simplifying this, we get:

k^2 - 16 = 0

Step 2: Solve for k
We can factor k^2 - 16 as (k + 4)(k - 4). Therefore, the solutions are k = 4 and k = -4.

However, we need to check if these solutions satisfy the original equation. Substituting k = 4, we get:

p(x) = (4^2 - 14)x^2 - 2x - 12 = -6x^2 - 2x - 12

The sum of the zeroes of this polynomial is:

a + b = -(-2)/(-6) = 1/3

This does not satisfy the given condition, so we reject k = 4 as a solution.

Substituting k = -4, we get:

p(x) = ((-4)^2 - 14)x^2 - 2x - 12 = -2x^2 - 2x - 12

The sum of the zeroes of this polynomial is:

a + b = -(-2)/(-2) = 1

Therefore, k = -4 is the solution that satisfies the given condition.

Answer: The value of k is -4.

Answer

+-4

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