Question

If both (x+2) and(2x-1) are factors of ax^2+2x+b then the value of a-b is

Answer 1

$Let \: the \: polynomial \:p(x) = ax^{2}+2x+b$

$Given \: (x+2) \:and \:(2x-1) \:are \:two \\ factors\: of \: p(x)$

$i ) If \: (x+2) \:is \: a \: factor \:p(x) \: then \\p(-2) = 0$

$a\times 2^{2} + 2\times (-2) + b = 0$

$\implies 4a - 4 + b = 0\: --(1)$

$ii ) If \: (2x-1) \:is \: a \: factor \:p(x) \: then \\p\big(\frac{1}{2}\big) = 0$

$a\times \big(\frac{1}{2}\big)^{2} + 2\times \frac{1}{2} + b = 0$

$\implies \frac{a}{4} + 1+ b = 0 \: --(2)$

/* Subtract equation (2) from equation (1), we get */

$4a - \frac{a}{4} - 5 = 0$

/* Multiplying each term by 4, we get */

$\implies 16a - a - 20 = 0$

$\implies 15a = 20$

$\implies a = \frac{20}{15}$

$\implies a = \frac{4}{3} \:--(3)$

/* Now , put value of a in equation (1) , we get */

$\implies 4\times \big(\frac{4}{3}\big) - 4 + b = 0$

$\implies \frac{16-12}{3} + b = 0$

$\implies \frac{4}{3} + b = 0$

$\implies b = \frac{-4}{3}$

Therefore.,

$\green {a = \frac{4}{3} \: and \: b = \frac{-4}{3} }$

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Answer 2

Answer:

Find x then solve the expression