Question

If alpha and beta are the zeroes of the polynomial p(x)=2x^2+3x+5, then find 1/alpha+1/beta

Answer 1

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QUESTION -

If alpha and beta are the zeroes of the polynomial p(x)=2x^2+3x+5, then find 1/alpha+1/beta.

SOLUTION :

From the above Question, we can gather the following information......

Alpha and beta are the zeroes of the polynomial p(x)=2x^2+3x+5.

So,

$p(x) = 2 { x } ^ 2 + 3x + 5 \\ \\ => p(x) = { x } ^ 2 + \dfrac{3}{2} x + \dfrac{5}{2}$

$\dfrac{1}{ \alpha } + \dfrac{ 1 }{ \beta } = \dfrac{ \alpha + \beta }{ \alpha \beta } = - \dfrac{ Sum \: Of \: Roots }{ Product \: Of Roots }$

Substuting the required values ;

$\dfrac{1}{ \alpha } + \dfrac { 1 }{ \beta } = - \dfrac{ \dfrac{3}{2} } { \dfrac{5}{2} } =- \dfrac{3}{5} ............... [ A ]$

Answer 2

Given that :

α and β are the zeros of the polynomial p(x) = 2x² + 3x + 5.

To find :-

The value of 1/α + 1/β.

Solution :-

We know,

Sum of the zeros of a quadratic polynomial(ax² + bx + c) is

α + β = -coefficient of x/coefficient of x² = -b/a

Product of the zeros of a quadratic polynomial(ax² + bx + c) is

αβ =  constant term/coefficient of x² = c/a

Now,

given polynomial is 2x² + 3x + 5.

• α + β = -b/a = -3/2

• αβ = c/a = 5/2

Calculating 1/α + 1/β :

= 1/α + 1/β

(Taking LCM)

= (α + β)/αβ

(Putting the known values)

= -3/2 ÷ 5/2

= -3/2 * 2/5

= -3/5