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If alpha and beta are the zeroes of the polynomial f(x)=x^2-3x 6, then find the value of 1/alpha 1/beta alpha^2 beta^2 -2 ×alpha×beta?
Most Upvoted Answer
If alpha and beta are the zeroes of the polynomial f(x)=x^2-3x 6, then...
Given:
The polynomial f(x) = x^2 - 3x + 6 has zeroes alpha and beta.
To find:
The value of (1/alpha) * (1/beta) * (alpha^2) * (beta^2) - 2 * alpha * beta.
Solution:
Step 1: Find the sum and product of the zeroes:
The sum of the zeroes is given by the formula: alpha + beta = -(-3) = 3.
The product of the zeroes is given by the formula: alpha * beta = 6.
Step 2: Find the value of (alpha^2) * (beta^2):
Using the identity (alpha + beta)^2 = (alpha^2) + 2 * alpha * beta + (beta^2), we can rewrite it as:
(3)^2 = (alpha^2) + 2 * alpha * beta + (beta^2).
Simplifying the equation:
9 = (alpha^2) + 2 * alpha * beta + (beta^2).
Substituting the value of (alpha * beta) from Step 1:
9 = (alpha^2) + 2 * 6 + (beta^2).
9 = (alpha^2) + 12 + (beta^2).
Rearranging the equation:
(alpha^2) + (beta^2) = 9 - 12.
(alpha^2) + (beta^2) = -3.
Step 3: Substitute the values in the given expression:
(1/alpha) * (1/beta) * (alpha^2) * (beta^2) - 2 * alpha * beta
= (1/alpha) * (1/beta) * (-3) - 2 * alpha * beta.
Simplifying the expression:
= (-3/alpha * beta) - 2 * alpha * beta.
Substituting the value of (alpha * beta) from Step 1:
= (-3/6) - 2 * 6.
= -1/2 - 12.
= -1/2 - 24/2.
= -25/2.
Final Answer:
The value of (1/alpha) * (1/beta) * (alpha^2) * (beta^2) - 2 * alpha * beta is -25/2.
The polynomial f(x) = x^2 - 3x + 6 has zeroes alpha and beta.
To find:
The value of (1/alpha) * (1/beta) * (alpha^2) * (beta^2) - 2 * alpha * beta.
Solution:
Step 1: Find the sum and product of the zeroes:
The sum of the zeroes is given by the formula: alpha + beta = -(-3) = 3.
The product of the zeroes is given by the formula: alpha * beta = 6.
Step 2: Find the value of (alpha^2) * (beta^2):
Using the identity (alpha + beta)^2 = (alpha^2) + 2 * alpha * beta + (beta^2), we can rewrite it as:
(3)^2 = (alpha^2) + 2 * alpha * beta + (beta^2).
Simplifying the equation:
9 = (alpha^2) + 2 * alpha * beta + (beta^2).
Substituting the value of (alpha * beta) from Step 1:
9 = (alpha^2) + 2 * 6 + (beta^2).
9 = (alpha^2) + 12 + (beta^2).
Rearranging the equation:
(alpha^2) + (beta^2) = 9 - 12.
(alpha^2) + (beta^2) = -3.
Step 3: Substitute the values in the given expression:
(1/alpha) * (1/beta) * (alpha^2) * (beta^2) - 2 * alpha * beta
= (1/alpha) * (1/beta) * (-3) - 2 * alpha * beta.
Simplifying the expression:
= (-3/alpha * beta) - 2 * alpha * beta.
Substituting the value of (alpha * beta) from Step 1:
= (-3/6) - 2 * 6.
= -1/2 - 12.
= -1/2 - 24/2.
= -25/2.
Final Answer:
The value of (1/alpha) * (1/beta) * (alpha^2) * (beta^2) - 2 * alpha * beta is -25/2.
Community Answer
If alpha and beta are the zeroes of the polynomial f(x)=x^2-3x 6, then...
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If alpha and beta are the zeroes of the polynomial f(x)=x^2-3x 6, then find the value of 1/alpha 1/beta alpha^2 beta^2 -2 ×alpha×beta? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If alpha and beta are the zeroes of the polynomial f(x)=x^2-3x 6, then find the value of 1/alpha 1/beta alpha^2 beta^2 -2 ×alpha×beta? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If alpha and beta are the zeroes of the polynomial f(x)=x^2-3x 6, then find the value of 1/alpha 1/beta alpha^2 beta^2 -2 ×alpha×beta?.
If alpha and beta are the zeroes of the polynomial f(x)=x^2-3x 6, then find the value of 1/alpha 1/beta alpha^2 beta^2 -2 ×alpha×beta? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If alpha and beta are the zeroes of the polynomial f(x)=x^2-3x 6, then find the value of 1/alpha 1/beta alpha^2 beta^2 -2 ×alpha×beta? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If alpha and beta are the zeroes of the polynomial f(x)=x^2-3x 6, then find the value of 1/alpha 1/beta alpha^2 beta^2 -2 ×alpha×beta?.
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