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Q. A quadratic equation ax² bx c = 0 is such that the sum of its roots is 5 less than the product of its roots. If both the roots are positive and one root is twice the other root, what will be the product of the roots?
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Q. A quadratic equation ax² bx c = 0 is such that the sum of its r...
Given:
Quadratic equation: ax² + bx + c = 0
Sum of roots = 5 less than the product of roots
Both roots are positive
One root is twice the other root
To Find:
Product of the roots of the quadratic equation
Solution:
Let the roots of the quadratic equation be p and 2p, where p is a positive number.
Step 1: Sum of the roots
The sum of the roots of a quadratic equation is given by the formula: Sum of roots = -b/a
In this case, the sum of the roots is 5 less than the product of the roots. So we can write the equation as: -b/a = 2p + p - 5
Simplifying the equation, we get: -b/a = 3p - 5
Step 2: Product of the roots
The product of the roots of a quadratic equation is given by the formula: Product of roots = c/a
In this case, the product of the roots is c/a = (2p)(p) = 2p²
Step 3: Equating the sum and product of roots
Equating the sum and product of the roots, we get: 3p - 5 = 2p²
Simplifying the equation, we get: 2p² - 3p + 5 = 0
Step 4: Solving the quadratic equation
Using the quadratic formula, p = (-b ± √(b² - 4ac)) / (2a), we can solve the equation 2p² - 3p + 5 = 0
The discriminant (b² - 4ac) is negative in this case, which means the equation has no real roots. However, we are given that both roots are positive, so we can conclude that the quadratic equation has complex roots.
Conclusion:
Since the quadratic equation has complex roots, we cannot directly find the product of the roots. Therefore, we cannot determine the product of the roots based on the information given.
Quadratic equation: ax² + bx + c = 0
Sum of roots = 5 less than the product of roots
Both roots are positive
One root is twice the other root
To Find:
Product of the roots of the quadratic equation
Solution:
Let the roots of the quadratic equation be p and 2p, where p is a positive number.
Step 1: Sum of the roots
The sum of the roots of a quadratic equation is given by the formula: Sum of roots = -b/a
In this case, the sum of the roots is 5 less than the product of the roots. So we can write the equation as: -b/a = 2p + p - 5
Simplifying the equation, we get: -b/a = 3p - 5
Step 2: Product of the roots
The product of the roots of a quadratic equation is given by the formula: Product of roots = c/a
In this case, the product of the roots is c/a = (2p)(p) = 2p²
Step 3: Equating the sum and product of roots
Equating the sum and product of the roots, we get: 3p - 5 = 2p²
Simplifying the equation, we get: 2p² - 3p + 5 = 0
Step 4: Solving the quadratic equation
Using the quadratic formula, p = (-b ± √(b² - 4ac)) / (2a), we can solve the equation 2p² - 3p + 5 = 0
The discriminant (b² - 4ac) is negative in this case, which means the equation has no real roots. However, we are given that both roots are positive, so we can conclude that the quadratic equation has complex roots.
Conclusion:
Since the quadratic equation has complex roots, we cannot directly find the product of the roots. Therefore, we cannot determine the product of the roots based on the information given.
Community Answer
Q. A quadratic equation ax² bx c = 0 is such that the sum of its r...
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Q. A quadratic equation ax² bx c = 0 is such that the sum of its roots is 5 less than the product of its roots. If both the roots are positive and one root is twice the other root, what will be the product of the roots?
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Q. A quadratic equation ax² bx c = 0 is such that the sum of its roots is 5 less than the product of its roots. If both the roots are positive and one root is twice the other root, what will be the product of the roots? for CAT 2023 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Q. A quadratic equation ax² bx c = 0 is such that the sum of its roots is 5 less than the product of its roots. If both the roots are positive and one root is twice the other root, what will be the product of the roots? covers all topics & solutions for CAT 2023 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Q. A quadratic equation ax² bx c = 0 is such that the sum of its roots is 5 less than the product of its roots. If both the roots are positive and one root is twice the other root, what will be the product of the roots?.
Q. A quadratic equation ax² bx c = 0 is such that the sum of its roots is 5 less than the product of its roots. If both the roots are positive and one root is twice the other root, what will be the product of the roots? for CAT 2023 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Q. A quadratic equation ax² bx c = 0 is such that the sum of its roots is 5 less than the product of its roots. If both the roots are positive and one root is twice the other root, what will be the product of the roots? covers all topics & solutions for CAT 2023 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Q. A quadratic equation ax² bx c = 0 is such that the sum of its roots is 5 less than the product of its roots. If both the roots are positive and one root is twice the other root, what will be the product of the roots?.
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