If the sum of the roots of kx^2 + 2x + 3k = 0 is equal to their product, then k = A 3/2 B 2/3 C 2/3 D 3/2
Question
If the sum of the roots of $kx^{2} + 2x + 3k = 0$ is equal to their product, then $k =$
A $\frac{3}{2}$
B $\frac{2}{3}$
C $\frac{2}{3}$
D $\frac{3}{2}$
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Answer
To solve the problem of finding ( k ) for which the sum of the roots of the quadratic equation ( kx^2 + 2x + 3k = 0 ) is equal to their product, follow these steps:

Identify the coefficients: Given the quadratic equation ( kx^2 + 2x + 3k = 0 ), we can compare it with the standard form ( ax^2 + bx + c = 0 ).
 ( a = k )
 ( b = 2 )
 ( c = 3k )

Use the properties of quadratic roots:
 The sum of the roots ( \alpha ) and ( \beta ) of a quadratic equation ( ax^2 + bx + c = 0 ) is given by ( \alpha + \beta = \frac{b}{a} ).
 The product of the roots is ( \alpha \cdot \beta = \frac{c}{a} ).

Calculate the sum and product:
 The sum of the roots is:
$$ \alpha + \beta = \frac{b}{a} = \frac{2}{k} $$
 The product of the roots is:
$$ \alpha \cdot \beta = \frac{c}{a} = \frac{3k}{k} = 3 $$

Set the sum equal to the product:
Because we are given that the sum of the roots equals their product, we set the equations equal to each other:
$$ \frac{2}{k} = 3 $$

Solve for ( k ):
 Multiply both sides by ( k ):
$$ 2 = 3k $$
 Divide both sides by 3:
$$ k = \frac{2}{3} $$
Thus, the value of ( k ) is [ \boxed{\frac{2}{3}} ].
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