Question :
- [tex] \tt \dfrac{x+2}{x-2} + \dfrac{x-2}{x+2} = \dfrac{5}{6}[/tex]
Answer :
- [tex] \large \underline{\boxed{\bf{x = \dfrac{\pm 2\sqrt{119}}{7}}}}[/tex]
Explanation :
[tex] \tt : \implies \dfrac{x+2}{x-2} + \dfrac{x-2}{x+2} = \dfrac{5}{6}[/tex]
[tex] \tt : \implies \dfrac{(x+2)(x+2) + (x-2)(x-2)}{(x-2)(x+2)} = \dfrac{5}{6}[/tex]
[tex] \tt : \implies \dfrac{(x+2)^{2} + (x-2)^{2}}{(x-2)(x+2)} = \dfrac{5}{6}[/tex]
Now, we know that :
- [tex] \large \underline{\boxed{\bf{(a+b)^{2} = a^{2} + b^{2}+ 2ab}}}[/tex]
- [tex] \large \underline{\boxed{\bf{(a-b)^{2} = a^{2} + b^{2} - 2ab}}}[/tex]
- [tex] \large \underline{\boxed{\bf{(a+b)(a-b) = a^{2} - b^{2}}}}[/tex]
[tex] \tt : \implies \dfrac{x^{2}+2^{2}+ 2 \times x \times 2 + x^{2}+2^{2} - 2 \times x \times 2 }{x^{2}-2^{2}} = \dfrac{5}{6}[/tex]
[tex] \tt : \implies \dfrac{x^{2}+ 4 + \cancel{4x} + x^{2}+ 4 - \cancel{4x}}{x^{2}-4} = \dfrac{5}{6}[/tex]
[tex] \tt : \implies \dfrac{x^{2} + x^{2} + 4 + 4}{x^{2}-4} = \dfrac{5}{6}[/tex]
[tex] \tt : \implies \dfrac{2x^{2} + 8}{x^{2}-4} = \dfrac{5}{6}[/tex]
By cross multiply :
[tex] \tt : \implies (2x^{2} + 8)6= 5(x^{2}-4)[/tex]
[tex] \tt : \implies 12x^{2} + 48 = 5x^{2}-20[/tex]
[tex] \tt : \implies 12x^{2} + 48 - 5x^{2} + 20 = 0[/tex]
[tex] \tt : \implies 7x^{2} + 68 = 0[/tex]
[tex] \tt : \implies 7x^{2} + 0x + 68 = 0[/tex]
Now, by comparing with ax² + bx + c = 0, we have :
By using quadratic formula :
[tex] \large \underline{\boxed{\bf{x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}}}}[/tex]
[tex] \tt : \implies x = \dfrac{-(0) \pm \sqrt{(0)^{2} - 4(7)(68)}}{2(7)}[/tex]
[tex] \tt : \implies x = \dfrac{0 \pm \sqrt{0 - 1904}}{14}[/tex]
[tex] \tt : \implies x = \dfrac{\pm \sqrt{- 1904}}{14}[/tex]
[tex] \tt : \implies x = \dfrac{\pm \sqrt{2\times 2\times 2\times 2\times 7\times 17}}{14}[/tex]
[tex] \tt : \implies x = \dfrac{\pm \cancel{2} \times 2\sqrt{7\times 17}}{\cancel{14}}[/tex]
[tex] \tt : \implies x = \dfrac{\pm2\sqrt{119}}{7}[/tex]
[tex] \large \underline{\boxed{\bf{x = \dfrac{\pm 2\sqrt{119}}{7}}}}[/tex]
Hence value of [tex] \bf x =\dfrac{\pm 2\sqrt{119}}{7}[/tex]
