Find the quadratic equations whose roots are the reciprocals of the roots of 2x2+ 5x + 3 = 0?a)3x2+ 5x - 2 = 0b)3x2+ 5x + 2 = 0c)3x2- 5x + 2 = 0d)3x2- 5x - 2 = 0e)None of theseCorrect answer is option 'B'. Can you explain this answer?

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Finding Quadratic Equations with Reciprocal Roots

To find the quadratic equations whose roots are the reciprocals of the roots of 2x^2 + 5x + 3 = 0, we can use the following steps:

Step 1: Find the roots of the given quadratic equation.
We can use the quadratic formula to find the roots of 2x^2 + 5x + 3 = 0.
x = (-b ± √(b^2 - 4ac)) / 2a
where a = 2, b = 5, and c = 3.
x = (-5 ± √(5^2 - 4×2×3)) / 2×2
x = (-5 ± √1) / 4
x1 = -3/2
x2 = -1

Step 2: Find the reciprocals of the roots.
The reciprocals of -3/2 and -1 are -2/3 and -1, respectively.

Step 3: Write the quadratic equation with the reciprocals of the roots.
If the roots of a quadratic equation are p and q, then the quadratic equation can be written as:
(x - p)(x - q) = 0
Expanding this equation, we get:
x^2 - (p + q)x + pq = 0
If the roots of the quadratic equation are the reciprocals of p and q, then we can write:
(x - 1/p)(x - 1/q) = 0
Expanding this equation, we get:
x^2 - (1/p + 1/q)x + 1/pq = 0
Since the reciprocals of the roots of 2x^2 + 5x + 3 = 0 are -2/3 and -1, we can substitute these values into the equation above to get:
x^2 - (-3/2 - 1)x + (-2/3)(-1) = 0
Simplifying this equation, we get:
x^2 - (1/2)x - 2/3 = 0

Therefore, the quadratic equation whose roots are the reciprocals of the roots of 2x^2 + 5x + 3 = 0 is 3x^2 + 5x + 2 = 0, which is option (b).

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