Question

write the standard form.of equation using factorization method,solve the following quadratic equation x-1/x-2+x-3/x-4=3 1/3;x is not equal to2,4​

Answer 1

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

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

$\rm :\longmapsto\:\dfrac{(x - 1)(x - 4) + (x - 3)(x - 2)}{(x - 2)(x - 4)} = \dfrac{10}{3}$

$\rm :\longmapsto\:\dfrac{{x}^{2} - x - 4x + 4 + {x}^{2} - 3x - 2x + 6 }{ {x}^{2} - 2x - 4x + 8 } = \dfrac{10}{3}$

$\rm :\longmapsto\:\dfrac{ {2x}^{2} - 10x + 10 }{ {x}^{2} - 6x + 8} = \dfrac{10}{3}$

$\rm :\longmapsto\: {10x}^{2} - 60x + 80 = {6x}^{2} - 30x + 30$

$\rm :\longmapsto\: {4x}^{2} - 30x + 50 = 0$

$\rm :\longmapsto\: {2x}^{2} - 15x + 25 = 0$

$\rm :\longmapsto\: {2x}^{2} - 10x - 5x + 25 = 0$

$\rm :\longmapsto\:2x(x - 5) - 5(x - 5) = 0$

$\rm :\longmapsto\:(x - 5)(2x - 5) = 0$

$\bf\implies \:x = 5 \: \: \: or \: \: \: x = \dfrac{5}{2}$

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