If one Zero of Polynomial (a^2 + 9) x^2 + 13x + 6a is Reciprocal of other, Find Value of a
By BYJU'S Exam Prep
Updated on: October 17th, 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.
Table of content
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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