Question

Solve this class 9 Question - ​

Answer 1

this is the answer
i hope this will help you

Answer 2

The value of x² + xy + y² is 195 when $x=\frac{\sqrt{3}-2}{\sqrt{3}+2}and\ y=\frac{\sqrt{3}+2}{\sqrt{3}-2}$.

Given :

$x=\frac{\sqrt{3}-2}{\sqrt{3}+2}and\ y=\frac{\sqrt{3}+2}{\sqrt{3}-2}$

To find :

$x^{2} +y^{2} + xy$

Formula used :

$(a + b)^{2} = a^{2} +2ab + b^{2} \\(a - b)^{2} = a^{2} - 2ab + b^{2} \\(a + b) (a - b) = a^{2} - b^{2}$

Solution:

Step 1: Rationalising the denominator of x :

$x=\frac{\sqrt{3}- 2}{\sqrt{3}+2} \times \frac{\sqrt{3} - 2}{\sqrt{3} - 2}$

$x=\frac{(\sqrt{3} - 2)^{2}}{(\sqrt{3})^{2} - 2^{2}} \\\\\ x =\frac{(\sqrt{3})^2 + 2^2-2 \sqrt{3} \times {2}}{3-2}\\\\\ x= \frac{3 + 4 - 2 \sqrt{3} \times 2}{3 - 4}$

[Formula - (a - b)² = a² -2ab + b²  and (a + b) (a - b) = a² - b²]

$x=\frac{7 - 4 \sqrt{3}}{-1}$

$x= - {7 + 4 \sqrt{3}$..........(1)

Step 2: Rationalising the denominator of y :

$y=\frac{\sqrt{3}+ 2}{\sqrt{3}-2} \times \frac{\sqrt{3} + 2}{\sqrt{3} +2}$

$y =\frac{(\sqrt{3}+{2})^{2}}{(\sqrt{3})^{2}-(2)^{2}}$

$y =\frac{(\sqrt{3})^2 + 2^2+2 \sqrt{3} \times {2}}{3-2}$

[Formula - (a + b)² = a² + 2ab + b²  and (a + b) (a - b) = a² - b²]

$y =\frac{3+4+2 \sqrt{3} \times {2}}{3-4}$

$y=\frac{7 + 4 \sqrt{3}}{-1}$

$y= - {7 - 4 \sqrt{3}$..........(2)

Step 3: Substituting the value from eq.1 and eq. 2 in x + y :

$x+y= -7+4 \sqrt{3} +( -7 + 4 \sqrt{3})\\\\\x+ y = - 7 + 4 \sqrt{3} - 7 - 4 \sqrt{3}$

$x+ y = - 7 - 7 + 4 \sqrt{3} - 4 \sqrt{3}$

x + y = - 14 ………(3)

Step 4: Substituting the value from eq.1 and eq. 2 in  xy :

[Formula - (a + b) (a - b) = a² - b²]

$\begin{array}{l}xy = (- 7 + 4 \sqrt{3})( - 7 - 4 \sqrt{3}) \\\\xy = (-7)^{2} - 4(\sqrt{3})^{2} \\\\xy = 49 - 16 \times 3\\\\xy = 49 - 48\\\\ xy = 1\end{array}$........(4)

Step 5: Substituting the value from eq.3 and eq. 4 in x² + xy + y²:

[Formula - (a + b)² = a² +2ab + b² ]

$x^{2}+x y+y^{2} =(x+y)^{2}-x y\\x^{2}+x y+y^{2} =(-14)^{2}-1\\x^{2}+x y+y^{2} =196 - 1$

x² + xy + y² = 195

Hence, the value of x² + xy + y² is 195.

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