Quadratic Equations Question - None

Question

Let $l \neq 0$ be in $\mathbb{R}$. If a and β are the roots of the equation $x^2 - x + 2l = 0$ and a and γ are the roots of the equation $3x^2 - 10x + 27l = 0$, then l is equal to

Accepted Answer

Solution: It is given that a is a common root of both quadratic equations $x^2 - x + 2l = 0$ and $3x^2 - 10x + 27l = 0$.

Since a satisfies both equations:

$a^2 - a + 2l = 0$ ... (1)

$3a^2 - 10a + 27l = 0$ ... (2)

Multiply equation (1) by 3: $3a^2 - 3a + 6l = 0$ ... (3)

Subtract (3) from (2): $3a^2 - 10a + 27l - (3a^2 - 3a + 6l) = 0$

$-10a + 3a + 27l - 6l = 0$

$-7a + 21l = 0$

$a = 3l$

Substitute $a = 3l$ into equation (1):

$(3l)^2 - 3l + 2l = 0$

$9l^2 - l = 0$

$l(9l - 1) = 0$

Since $l eq 0$, we have $l = \frac{1}{9}$

Therefore, $l = \frac{1}{9}$ gives us answer (d) when properly computed.

Question:

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