Solve 3/x-2+4/x=0.0775

$$\frac{ 3 }{ x-2 } + \frac{ 4 }{ x } = 0.0775$$

Solve for x

$$x = \frac{\sqrt{1948561} + 1431}{31} \approx 91.190602085$$

$$x = \frac{1431 - \sqrt{1948561}}{31} \approx 1.13197856$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,2 since division by zero is not defined. Multiply both sides of the equation by x\left(x-2\right), the least common multiple of x-2,x.

$$x\times 3+\left(x-2\right)\times 4=0.0775x\left(x-2\right)$$

Use the distributive property to multiply x-2 by 4.

$$x\times 3+4x-8=0.0775x\left(x-2\right)$$

Combine x\times 3 and 4x to get 7x.

$$7x-8=0.0775x\left(x-2\right)$$

Use the distributive property to multiply 0.0775x by x-2.

$$7x-8=0.0775x^{2}-0.155x$$

Subtract 0.0775x^{2} from both sides.

$$7x-8-0.0775x^{2}=-0.155x$$

Add 0.155x to both sides.

$$7x-8-0.0775x^{2}+0.155x=0$$

Combine 7x and 0.155x to get 7.155x.

$$7.155x-8-0.0775x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-0.0775x^{2}+7.155x-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.0775 for a, 7.155 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7.155±\sqrt{7.155^{2}-4\left(-0.0775\right)\left(-8\right)}}{2\left(-0.0775\right)}$$

Square 7.155 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-7.155±\sqrt{51.194025-4\left(-0.0775\right)\left(-8\right)}}{2\left(-0.0775\right)}$$

Multiply -4 times -0.0775.

$$x=\frac{-7.155±\sqrt{51.194025+0.31\left(-8\right)}}{2\left(-0.0775\right)}$$

Multiply 0.31 times -8.

$$x=\frac{-7.155±\sqrt{51.194025-2.48}}{2\left(-0.0775\right)}$$

Add 51.194025 to -2.48 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-7.155±\sqrt{48.714025}}{2\left(-0.0775\right)}$$

Take the square root of 48.714025.

$$x=\frac{-7.155±\frac{\sqrt{1948561}}{200}}{2\left(-0.0775\right)}$$

Multiply 2 times -0.0775.

$$x=\frac{-7.155±\frac{\sqrt{1948561}}{200}}{-0.155}$$

Now solve the equation x=\frac{-7.155±\frac{\sqrt{1948561}}{200}}{-0.155} when ± is plus. Add -7.155 to \frac{\sqrt{1948561}}{200}.

$$x=\frac{\sqrt{1948561}-1431}{-0.155\times 200}$$

Divide \frac{-1431+\sqrt{1948561}}{200} by -0.155 by multiplying \frac{-1431+\sqrt{1948561}}{200} by the reciprocal of -0.155.

$$x=\frac{1431-\sqrt{1948561}}{31}$$

Now solve the equation x=\frac{-7.155±\frac{\sqrt{1948561}}{200}}{-0.155} when ± is minus. Subtract \frac{\sqrt{1948561}}{200} from -7.155.

$$x=\frac{-\sqrt{1948561}-1431}{-0.155\times 200}$$

Divide \frac{-1431-\sqrt{1948561}}{200} by -0.155 by multiplying \frac{-1431-\sqrt{1948561}}{200} by the reciprocal of -0.155.

$$x=\frac{\sqrt{1948561}+1431}{31}$$

The equation is now solved.

$$x=\frac{1431-\sqrt{1948561}}{31} x=\frac{\sqrt{1948561}+1431}{31}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,2 since division by zero is not defined. Multiply both sides of the equation by x\left(x-2\right), the least common multiple of x-2,x.

$$x\times 3+\left(x-2\right)\times 4=0.0775x\left(x-2\right)$$

Use the distributive property to multiply x-2 by 4.

$$x\times 3+4x-8=0.0775x\left(x-2\right)$$

Combine x\times 3 and 4x to get 7x.

$$7x-8=0.0775x\left(x-2\right)$$

Use the distributive property to multiply 0.0775x by x-2.

$$7x-8=0.0775x^{2}-0.155x$$

Subtract 0.0775x^{2} from both sides.

$$7x-8-0.0775x^{2}=-0.155x$$

Add 0.155x to both sides.

$$7x-8-0.0775x^{2}+0.155x=0$$

Combine 7x and 0.155x to get 7.155x.

$$7.155x-8-0.0775x^{2}=0$$

Add 8 to both sides. Anything plus zero gives itself.

$$7.155x-0.0775x^{2}=8$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-0.0775x^{2}+7.155x=8$$

Divide both sides of the equation by -0.0775, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-0.0775x^{2}+7.155x}{-0.0775}=\frac{8}{-0.0775}$$

Dividing by -0.0775 undoes the multiplication by -0.0775.

$$x^{2}+\frac{7.155}{-0.0775}x=\frac{8}{-0.0775}$$

Divide 7.155 by -0.0775 by multiplying 7.155 by the reciprocal of -0.0775.

$$x^{2}-\frac{2862}{31}x=\frac{8}{-0.0775}$$

Divide 8 by -0.0775 by multiplying 8 by the reciprocal of -0.0775.

$$x^{2}-\frac{2862}{31}x=-\frac{3200}{31}$$

Divide -\frac{2862}{31}, the coefficient of the x term, by 2 to get -\frac{1431}{31}. Then add the square of -\frac{1431}{31} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2862}{31}x+\left(-\frac{1431}{31}\right)^{2}=-\frac{3200}{31}+\left(-\frac{1431}{31}\right)^{2}$$

Square -\frac{1431}{31} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2862}{31}x+\frac{2047761}{961}=-\frac{3200}{31}+\frac{2047761}{961}$$

Add -\frac{3200}{31} to \frac{2047761}{961} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{2862}{31}x+\frac{2047761}{961}=\frac{1948561}{961}$$

Factor x^{2}-\frac{2862}{31}x+\frac{2047761}{961}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1431}{31}\right)^{2}=\frac{1948561}{961}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1431}{31}\right)^{2}}=\sqrt{\frac{1948561}{961}}$$

Simplify.

$$x-\frac{1431}{31}=\frac{\sqrt{1948561}}{31} x-\frac{1431}{31}=-\frac{\sqrt{1948561}}{31}$$

Add \frac{1431}{31} to both sides of the equation.

$$x=\frac{\sqrt{1948561}+1431}{31} x=\frac{1431-\sqrt{1948561}}{31}$$