Find the value of $a$ and $b$ for the following algebric expressions.
$a z^{2} + 10 z - 75 = (z + b) (5 z - 15)$

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Solution

$a z^{2} - 10 z - 75 = (z + b) (5 z - 15)$

or, $a z^{2} - 10 z - 75 = 5 z^{2} + 5 b z - 15 z - 15 b$ .

or $a z^{2} - 10 z - 75 = 5 z^{2} - (15 - 5 b) z - 15 b$ .

Equating both sides,

$a = 5$

$10 = 15 - 5 b$ or, $75 = 15 b$ .

or, $5 b = 5$ or, $b = \frac{75}{15}$

$∴ b = 1$ $∴ b = 5$
Ans $: a = 5, b = 1, 5$

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