Question

x-2/x-4 +x-6/x-8 = 20/3​

Answer 1

$\large\underline{\sf{Solution-}}$

$\rm :\longmapsto\:\dfrac{x - 2}{x - 4} + \dfrac{x - 6}{x - 8} = \dfrac{20}{3}$

$\rm :\longmapsto\:\dfrac{x - 4 + 2}{x - 4} + \dfrac{x - 8 + 2}{x - 8} = \dfrac{20}{3}$

$\rm :\longmapsto\:\dfrac{x - 4}{x - 4} + \dfrac{2}{x - 4} + \dfrac{2}{x - 8} + \dfrac{x - 8}{x - 8} = \dfrac{20}{3}$

$\rm :\longmapsto\:1 + \dfrac{2}{x - 4} + 1 + \dfrac{2}{x - 8} = \dfrac{20}{3}$

$\rm :\longmapsto\:\dfrac{2}{x - 4} + \dfrac{2}{x - 8} = \dfrac{20}{3} - 2$

$\rm :\longmapsto\:\dfrac{2}{x - 4} + \dfrac{2}{x - 8} = \dfrac{20 - 6}{3}$

$\rm :\longmapsto\:\dfrac{2}{x - 4} + \dfrac{2}{x - 8} = \dfrac{14}{3}$

$\rm :\longmapsto\:\dfrac{1}{x - 4} + \dfrac{1}{x - 8} = \dfrac{7}{3}$

$\rm :\longmapsto\:\dfrac{x - 8 + x - 4}{(x - 4)(x - 8)} = \dfrac{7}{3}$

$\rm :\longmapsto\:\dfrac{2x - 12}{ {x}^{2} - 4x - 8x + 32 } = \dfrac{7}{3}$

$\rm :\longmapsto\:\dfrac{2x - 12}{ {x}^{2} - 12x + 32 } = \dfrac{7}{3}$

$\rm :\longmapsto\: {7x}^{2} - 84x + 224 = 6x - 36$

$\rm :\longmapsto\: {7x}^{2} - 90x + 260 =0$

Using Quadratic formula, the values of 'x' is given by

$\rm :\longmapsto\:x = \dfrac{ - b \: \pm \: \sqrt{ {b}^{2} - 4ac}}{2a}$

Here,

• a = 7

• b = - 90

• c = 260

On substituting the values in above formula, we get

$\rm :\longmapsto\:x = \dfrac{ - ( - 90) \: \pm \: \sqrt{ {( - 90)}^{2} - 4 \times 7 \times 260}}{2 \times 7}$

$\rm :\longmapsto\:x = \dfrac{ 90\: \pm \: \sqrt{ 8100 - 7280}}{14}$

$\rm :\longmapsto\:x = \dfrac{ 90\: \pm \: \sqrt{ 820}}{14}$

$\rm :\longmapsto\:x = \dfrac{ 90\: \pm \: \sqrt{ 2 \times 2 \times 205}}{14}$

$\rm :\longmapsto\:x = \dfrac{ 90\: \pm \: 2\sqrt{205}}{14}$

$\rm :\longmapsto\:x = \dfrac{ 45\: \pm \: \sqrt{205}}{7}$

$\rm \: \therefore\:\boxed{ \red{ \bf \: x = \dfrac{45+\sqrt{205} }{7}}} \: \: \: or \: \: \: \boxed{ \red{ \bf \: x = \dfrac{45 - \sqrt{205}}{7}}}$

Additional Information :-

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

• If Discriminant, D > 0, then roots of the equation are real and unequal.

• If Discriminant, D = 0, then roots of the equation are real and equal.

• If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

• Discriminant, D = b² - 4ac