If one Zero of Polynomial (a^2 + 9) x^2 + 13x + 6a is Reciprocal of other, Find Value of a

By Mohit Uniyal|Updated : May 15th, 2023

If one zero of polynomial (a^2 + 9) x^2 + 13x + 6a is reciprocal of other, find value of a.

To find the value of "a," when one zero of the polynomial (a^2 + 9) x^2 + 13x + 6a is the reciprocal of the other, you can follow these steps:

Step 1: Set up the equation using the given information. Let the two zeros of the polynomial be α and β, with α being the reciprocal of β. So, we have the following relationship: α * β = 1.

Step 2: Apply Vieta's formulas. According to Vieta's formulas, the sum of the zeros (α + β) is equal to the negation of the coefficient of the linear term (x term) divided by the coefficient of the quadratic term (x^2 term).

Step 3: Express α and β in terms of a, which is α = 1/β.

Step 4: Substitute the expressions for α and β into the equation. Replace α with 1/β in the equation α + β = -13 / (a^2 + 9): 1/β + β = -13 / (a^2 + 9).

Step 5: Multiply both sides of the equation by β(a^2 + 9) to eliminate the denominators. (a^2 + 9) + β^2 = -13β.

Step 6: Rearrange the equation to a quadratic form. Rearrange the equation: β^2 + 13β + (a^2 + 9) = 0.

Step 7: Solve the quadratic equation.

Step 8: Find the value of a.

If one Zero of Polynomial (a^2 + 9) x^2 + 13x + 6a is Reciprocal of other, Find Value of a

We are given the polynomial (a^2 + 9) x^2 + 13x + 6a, with roots p and 1/p. We need to find the value of a.

First, let's consider the product of the roots:

Product of the roots = p * (1/p) = 1

We know that the product of the roots is equal to the constant term (6a) divided by the coefficient of x^2, which is (a^2 + 9).

So, we have:

(6a) / (a^2 + 9) = 1

To solve this equation, we can multiply both sides by (a^2 + 9):

6a = a^2 + 9

Rearranging the terms, we get a quadratic equation:

a^2 - 6a + 9 = 0

This quadratic equation can be factored as a perfect square:

(a - 3)^2 = 0

Taking the square root of both sides, we have:

a - 3 = 0

Solving for a, we get:

a = 3

Therefore, the value of a that satisfies the given conditions is 3.

Answer

If one zero of polynomial (a^2 + 9) x^2 + 13x + 6a is reciprocal of other, then a = 3.

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