Question

If alpha and beta are the zeroes of the polynomial 2x^2+7x+5 then find the value of alpha+beta+alpha×beta

Answer 1

Compare the given quadratic equation with its standard form ax^2 + bx + c = 0
a = 2, b = 7 and c = 5
Alpha + beta = -b/a
= -7/2

Alpha x beta = c/a
= 5/2

Alpha + beta + alpha x beta
= (-7/2) + 5/2
= (-7+5)/2
= -2/2
= -1

Answer 2

Answer:-

$\boxed{ \alpha + \beta + \alpha \beta = - 1} \:$

Explanation :-

Formula used :-

$\small \: \star \: sum \: of \: zeros = \frac{ - coeff. \: of \: x}{coeff. \: of \: {x}^{2} } \\ \\ \star \: \small \: product \: of \: zeros \: = \frac{constant}{coeff. \: of \: {x}^{2} } \\ \\ \boxed{ here \: coeff. = coefficient}$

Solution :-

According to the question,

$\alpha \: and \: \beta \: \: are \: the \: zeros \: of \: \\ 2 {x}^{2} + 7x + 5 \: = 0 \\ \\ \therefore \\ \\ \implies \: \small \alpha + \beta = \frac{ - 7}{2} (sum \: of \: zeros)\\ \\ \small \implies \: \alpha \beta = \frac{5}{2} (product \: of \: zeros)$

We find ,

$\implies \: \alpha + \beta + \alpha \beta$

Substitute the values,

$\implies \: \frac{ - 7}{2} + \frac{5}{2} \\ \\ \implies \: \frac{ - 7 + 5}{2} \\ \\ \implies \: \frac{ - 2}{2} \\ \\ \implies \: - 1 \\ \\ therefore \\ \\ \implies \: \red{\boxed{ \alpha + \beta + \alpha \beta = - 1}}$