Question

Which equation has the solutions x= 1 =+/- sqrt5?

Answer 1

$Form \: of \: quadratic \: equation \: whose \\solutions \: are \: \alpha \:and \:\beta \: is \\x^{2} - (\alpha + \beta)x + \alpha \beta = 0$

$Here, \alpha = 1 + \sqrt{5} , \: and \: \beta = 1-\sqrt{5}$

$\alpha + \beta\\ = 1+\sqrt{5} + 1 - \sqrt{5} \\= 2$

$\alpha \cdot \beta \\= (1+\sqrt{5} )( 1 - \sqrt{5}) \\= 1^{2} - \sqrt{5}^{2} \\= 1 - 5 \\= -4$

Therefore.,

$\pink { Required \: Quadratic \: eqution ,}$

$x^{2} - 2x + (-4) = 0$

$\implies x^{2} - 2x - 4 = 0$

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