Question

IF a=3+√7/ 3-√7 and b= 3-√7/3+√7 find the value of a²+b²

Answer 1

Answer:

a²+b²=30

Step-by-step explanation:

Multiply a & b with their conjugates

$a=\frac{3+\sqrt{7} }{3-\sqrt{7} }  * \frac{3+\sqrt{7} }{3+\sqrt{7} } =\frac{(3+\sqrt{7} )^{2} }{3^{2}-7} \\ \\ \\=\frac{9+2\sqrt{7}+7 }{2} =\frac{16+2\sqrt{7} }{2} =\frac{2(8+\sqrt{7}) }{2} \\ =8+\sqrt{7}$

Similarly

$b=\frac{3-\sqrt{7} }{3+\sqrt{7} }  * \frac{3-\sqrt{7} }{3-\sqrt{7} } = \frac{(3-\sqrt{7} )^{2} }{3^{2} -\sqrt{7} ^{2} } \\ \\ =\frac{9-2\sqrt{7}+7 }{9-7} =\frac{16-2\sqrt{7} }{2} \\ =8-\sqrt{7}$

$a^{2} + b^{2} = (8+\sqrt{7}) ^{2} +(8-\sqrt{7} )^{2}\\    = 8+2\sqrt{7} +7+8-2\sqrt{7} +7\\=30$