Question

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

Answer 1

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