Question

√2-1/√2+1=a+b√2.find a and b.​

Answer 1

Answer :

a = 3, b = -2

Step-by-step explanation :

$\underline{\underline{\textbf{Rationalizing factor :}}}$

$\bigstar$    The factor of multiplication by which rationalization is done, is called as rationalizing factor.

$\bigstar$    If the product of two surds is a rational number, then each surd is a rationalizing factor to other.

$\bigstar$    For example, rationalizing factor of (3 + √2) is (3 - √2)

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$\bf{\frac{\sqrt{2}-1}{\sqrt{2}+1}=a+b\sqrt{2}$

Rationalizing factor = √2 - 1

$\bf=\frac{\sqrt{2}-1}{\sqrt{2}+1} \times\frac{\sqrt{2}-1}{\sqrt{2}-1} \\\\\\ = \frac{(\sqrt{2}-1)(\sqrt{2}-1)}{(\sqrt{2}+1)(\sqrt{2}-1)} \\\\\\ = \frac{\sqrt{2}(\sqrt{2}-1)-1(\sqrt{2}-1)}{\sqrt{2}^2-1^2} \\\\\\ =\frac{\sqrt{2}^2-\sqrt{2}-\sqrt{2}+1}{2-1} \\\\\\ =\frac{2-2\sqrt{2}+1}{1} \\\\\\ =3-2\sqrt{2} \\\\\\=3+(-2)\sqrt{2} \\\\\\=a+b\sqrt{2} \\\\\\ \longrightarrow a=3,b=-2$

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Formula Used :

(x + y) (x - y) = x² - y²

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