Question

(x + 3 / x -2) - (1-x / x) = 17/4. Solve step by step.

Answer 1

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

$\rm :\longmapsto\:\dfrac{x + 3}{x - 2} - \dfrac{1 - x}{x} = \dfrac{17}{4}$

$\rm :\longmapsto\:\dfrac{(x + 3)x - (x - 2)(1 - x)}{x(x - 2)} = \dfrac{17}{4}$

$\rm :\longmapsto\:\dfrac{ {x}^{2} + 3x - ( x - {x}^{2} - 2 + 2x )}{x(x - 2)} = \dfrac{17}{4}$

$\rm :\longmapsto\:\dfrac{ {x}^{2} + 3x - ( 3x - {x}^{2} - 2)}{x(x - 2)} = \dfrac{17}{4}$

$\rm :\longmapsto\:\dfrac{ {x}^{2} + 3x - 3x + {x}^{2} + 2}{x(x - 2)} = \dfrac{17}{4}$

$\rm :\longmapsto\:\dfrac{ 2{x}^{2} + 2}{ {x}^{2} - 2x} = \dfrac{17}{4}$

$\rm :\longmapsto\:17 {x}^{2} - 34x = {8x}^{2} + 8$

$\rm :\longmapsto\:9{x}^{2} - 34x - 8 = 0$

$\rm :\longmapsto\:9{x}^{2} - 36x + 2x - 8 = 0$

$\rm :\longmapsto\:9x(x - 4) + 2(x - 4) = 0$

$\rm :\longmapsto\:(x - 4)(9x + 2) = 0$

$\bf\implies \:x = 4 \: \: \: or \: \: \: x = - \dfrac{2}{9}$

Additional Information :-

There are 4 methods by which you can solve a quadratic equation such as:

• Method of factorization,

• Method of completing the square,

• Method of quadratic formula,

and

• Method of plotting graph

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