Question

If α and β are the zeros of the polynomial xˆ2 + x - 2. Find the value of 1/α - 1/β​

Answer 1

Answer

¹/α + ¹/β = ¹/₂

Given

α , β are zeroes of the polynomial x² + x - 2

To Find

$\rm Value\ of\ \dfrac{1}{\alpha}+\dfrac{1}{\beta}$

Concept Used

For quadratic polynomial ,

Sum of zeroes ,

$\to\ \rm \alpha+\beta=\dfrac{-\ coefficient\ of\ x}{coefficient\ of\ x^2}\\\\\to\ \rm \alpha+\beta=\dfrac{-b}{a}$

Product of zeroes ,

$\to\ \rm \alpha \beta=\dfrac{constant\ term}{coefficient\ of\ x^2}\\\\\to\ \rm \alpha \beta=\dfrac{c}{a}$

Solution

Compare given quadratic equation x² + x - 2 with ax² + bx + c , we get ,

• a = 1 , b = 1 , c = -2

Sum of zeroes ,

$\to\ \rm \alpha +\beta =\dfrac{-1}{1}\\\\\to\ \rm \alpha+\beta=-1\ ...(1)$

Product of zeroes ,

$\to\ \rm \alpha \beta =\dfrac{-2}{1}\\\\\to\ \rm \alpha \beta=-2\ ...(2)$

Now , we need to find the value of ,

$\rm \dfrac{1}{\alpha}+\dfrac{1}{\beta}\\\\\to\ \rm \dfrac{\beta +\alpha}{\alpha \beta}\\\\\to\ \rm \dfrac{(\alpha +\beta)}{(\alpha \beta)}\\\\\to\ \rm \dfrac{(-1)}{(-2)}\\\\\leadsto\ \rm{\green{\dfrac{1}{2}\ \; \bigstar}}$