Question

if alpha and beta are the zeros of a quadratic polynomial x square - 2 X + 3 find a polynomial whose roots are 1. alpha + beta, beta + Alpha 2.alpha-1/alpha+1,beta-1/beta+1​

Answer 1

$\huge{\underline{\boxed{\bf{\blue{Answer:-}}}}}$

$\boxed{\sf{x^{2} - 6x + 11 = 0}}$

$\huge{\underline{\boxed{\bf{\blue{Explainationtion:-}}}}}$

Let α, β be the roots of the equation

x² - 2x + 3 = 0

Let f(x) = x² - 2x + 3

★ Given:-

• Roots are increased by 2, so new roots of the equation are α + 2, β + 2.
• Let y = α + 2 which is the required root of new

So, α = y - 2

But, we know α is root of f(x), therefore

f(α) = 0

But α = y - 2, so

f(y - 2) = 0

(y - 2)² - 2(y - 2) + 3 = 0

y² - 6y + 11 = 0,

★ Point to remember:-

Since similar argument hold for other root as well, hence this equation represents the one with roots α + 2 and β + 2.

Changing the variable y to x, we get,

x² - 6x + 11 = 0, which is the required equation.

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