Consider the following2x2 + 3x - α- 0 has roots -2 and βwhile the equation x2 - 3mx + 2m2 = 0 has both roots positive, where α 0 and β 0. 39.Q. What is the value of α?a)1/2b)1c)2d)4Correct answer is option 'C'. Can you explain this answer?

Question

Consider the following2x2 + 3x - &alpha;- 0 has roots -2 and &beta;while the equation x2 - 3mx + 2m2 = 0 has both roots positive, where &alpha;> 0 and &beta; > 0. 39.Q. What is the value of &alpha;?a)1/2b)1c)2d)4Correct answer is option 'C'. Can you explain this answer?

Answer

Given Information:
The quadratic equation 2x2 + 3x - α = 0 has roots -2 and β.
The quadratic equation x2 - 3mx + 2m2 = 0 has both roots positive.

Solution:

Step 1: Finding the roots of the first equation
Given that the quadratic equation 2x2 + 3x - α = 0 has roots -2 and β.
The sum of the roots of a quadratic equation ax2 + bx + c = 0 is given by -b/a.
So, the sum of the roots of the equation 2x2 + 3x - α = 0 is -3/2.
Therefore, -2 + β = -3/2
β = -3/2 + 2
β = 1/2

Step 2: Finding the value of α
The product of the roots of a quadratic equation ax2 + bx + c = 0 is given by c/a.
So, the product of the roots of the equation 2x2 + 3x - α = 0 is -α/2.
Since one root is -2 and the other root is β = 1/2, we have:
-2 * 1/2 = -α/2
-1 = -α/2
α = 2 * 1
α = 2
Therefore, the value of α is 2.
So, the correct answer is option c) 2.

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