Question

Solve the following equation and also check your result:
[(2x + 3) + (x + 5)]² + [(2x + 3) − (x + 5)]² = 10x² + 92

Answer 1

\[[(2x + 3) + (x + 5) ]^2 + [(2x + 3) - (x + 5) ]^2 = 10 x^2 + 92\]
\[\text{ or }(3x + 8 )^2 + (x - 2 )^2 = 10 x^2 + 92\]
\[\text{ or }9 x^2 + 48x + 64 + x^2 - 4x + 4 = 10 x^2 + 92 [ (a + b )^2 = a^2 + b^2 + 2ab and (a - b )^2 = a^2 + b^2 - 2ab ]\]
\[\text{ or }10 x^2 - 10 x^2 + 44x = 92 - 68\]
\[\text{ or }x = \frac{24}{44}\]
\[\text{ or }x = \frac{6}{11}\]
\[\text{ Thus, }x = \frac{6}{11}\text{ is the solution of the given equation . }\]
\[\text{ Check: }\]
\[\text{ Substituting }x = \frac{6}{11}\text{ in the given equation, we get: }\]
\[\text{ L . H . S . }= \left[ \left( 2 \times \frac{6}{11} + 3 \right) + \left( \frac{6}{11} + 5 \right) \right]^2 + \left[ \left( 2 \times \frac{6}{11} + 3 \right) - \left( \frac{6}{11} + 5 \right) \right]^2 \]
\[ = \left[ \left( \frac{45}{11} \right) + \left( \frac{61}{11} \right) \right]^2 + \left[ \left( \frac{45}{11} \right) - \left( \frac{61}{11} \right) \right]^2 \]
\[ = \left( \frac{106}{11} \right)^2 + \left( \frac{- 16}{11} \right)^2 \]
\[ = \frac{11492}{121}\]
\[\text{ R . H . S . }= 10 \times \left( \frac{6}{11} \right)^2 + 92 = \frac{360}{121} + 92 = \frac{11492}{121}\]
\[ \therefore\text{ L . H . S . = R . H . S . for }x = \frac{6}{11}\]