Question

If alpha=√2 and beta =√5 what is the quadratic equation

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{Quadratic\:eqn\to x^{2}-(\sqrt{2}+\sqrt{5})x+\sqrt{10}=0}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given : }} \\ \tt: \implies \alpha = \sqrt{2} \\ \\ \tt: \implies \beta = \sqrt{5} \\ \\ \red{\underline \bold{To \: Find : }} \\ \tt: \implies Quadratic \: eqn = ?$

• According to given question :

$\bold{For \: sum \: of \: zeroes} \\ \green{\tt: \implies \alpha + \beta = \sqrt{2} + \sqrt{5} } \\ \\ \bold{For \: product \: of \: zeroes} \\ \tt: \implies \alpha \beta = \sqrt{2} \sqrt{5} \\ \\ \green{\tt: \implies \alpha \beta = \sqrt{10} } \\ \\ \bold{For \: quadratic \: eqn : } \\ \tt: \implies {x}^{2} - (sum \: of \: zeroes)x + (product \: of \: zeroes) = 0 \\ \\ \tt: \implies {x}^{2} - ( \alpha + \beta )x + \alpha \beta = 0 \\ \\ \green{\tt: \implies {x}^{2} - ( \sqrt{2} + \sqrt{5} )x + \sqrt{10} = 0}$

Answer 2

GIVEN :-

❈ α = √2

❈ β = √5

TO FIND :-

☛ The quadratic equation whose zeros are α and β.

SOLUTION :-

Sum of the zeros :

S = α + β

= √2 + √5

Product of the zeros :

P = √2 × √5

= √10

We know that, a quadratic equation is in the following form :-

✏ x² - (sum of the zeros)x + (product of the zeros) = 0

✏ x² - Sx + P = 0

So, the required quadratic equation is :

x² - (√2 + √5)x + √10 = 0                  ...$\boxed{\huge\tt {\color{orange} {ANSWER}}}$