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Quadratic Formula

If α , β ar...

Question

If $α, β$ are of quadratic polynomial $2 x^{2} + 5 x + k$ , find the value of $k$ , such that $(α + β)^{2} - α \cdot β = 24$

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Solution

$α + β = - \frac{b}{a}$
$α β = \frac{c}{a}$
Given, $2 x^{2} + 5 x + k = 0 \to a = 2, b = 5, c = k$
or, $(α + β)^{2} - α β = 24$
or, ${(- \frac{b}{a})}^{2} - \frac{c}{a} = 24$
or, $b^{2} - c a = 24 a^{2}$
or, $5^{2} - 2 k = 24 (2)^{2}$
or, $2 k = 25 - 96 = - 71$
or, $∴ k = - \frac{71}{2}$

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