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If 3p^2 = 5p+2 and 3q^2 = 5q+2 where p!=...

If `3p^2 = 5p+2` and `3q^2 = 5q+2` where `p!=q, pq` is equal to

A

`2/3`

B

`-2/3`

C

`3/2`

D

`-3/2`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we start with the equations given: 1. \( 3p^2 = 5p + 2 \) 2. \( 3q^2 = 5q + 2 \) ### Step 1: Rearranging the equations We can rearrange both equations to bring all terms to one side: For the first equation: \[ 3p^2 - 5p - 2 = 0 \] For the second equation: \[ 3q^2 - 5q - 2 = 0 \] ### Step 2: Identifying the quadratic equation Both equations represent the same quadratic equation: \[ 3x^2 - 5x - 2 = 0 \] where \( p \) and \( q \) are the roots of this equation. ### Step 3: Using Vieta's formulas According to Vieta's formulas, for a quadratic equation of the form \( ax^2 + bx + c = 0 \): - The sum of the roots \( p + q \) is given by: \[ p + q = -\frac{b}{a} = -\frac{-5}{3} = \frac{5}{3} \] - The product of the roots \( pq \) is given by: \[ pq = \frac{c}{a} = \frac{-2}{3} \] ### Final Answer Thus, the value of \( pq \) is: \[ pq = -\frac{2}{3} \] ---
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