Question

if alpha and beta are the zeroes of equation x²-6x+a=0 which satisfies the relation 3alpha+2beta=16 then find a​

Answer 1

✪ᴀɴsᴡᴇʀ✪

• $\large{\boxed{\star \star \green{a = 8} \star \star }}$

☆ɢɪᴠᴇɴ☆

• $\alpha, \beta$ are roots of the equation

$\:\:\:\:\:\:\:\:\:\:\:\: x^2 - 6x + a = 0$

• $3\alpha + 2\beta = 16$

★ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴ★

Using relations between roots and co-efficient,

$\to \alpha + \beta = - \frac{coeff\:of\:x} {coeff\:of\:x^2}$

$\to \alpha + \beta = - \frac{-6} {1}$

$\to \alpha + \beta = 6..... (1)$

It is given that

$\:\:\:\:\:3\alpha + 2\beta = 16$

$\to \alpha + 2(\alpha + \beta) = 16$

$\to \alpha + 2\times 6= 16$[using (1)]

$\to \alpha = 16 - 12$

$\to \alpha = 4$

Since $\alpha$ is a root,

$\:\:\:\:\:\alpha ^2 - 6\alpha + a = 0$

$\to 4^2 - 6.4 + a = 0$

$\to 16 - 24 + a = 0$

$\to - 8 + a = 0$

$\to a = 8$