Solve 2/x-5/x-1=7

$$\frac { 2 } { x } - \frac { 5 } { x - 1 } = 7$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{10}i+2}{7}\approx 0.285714286-0.451753951i$$

$$x=\frac{2+\sqrt{10}i}{7}\approx 0.285714286+0.451753951i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,1 since division by zero is not defined. Multiply both sides of the equation by x\left(x-1\right), the least common multiple of x,x-1.

$$\left(x-1\right)\times 2-x\times 5=7x\left(x-1\right)$$

Use the distributive property to multiply x-1 by 2.

$$2x-2-x\times 5=7x\left(x-1\right)$$

Use the distributive property to multiply 7x by x-1.

$$2x-2-x\times 5=7x^{2}-7x$$

Subtract 7x^{2} from both sides.

$$2x-2-x\times 5-7x^{2}=-7x$$

Add 7x to both sides.

$$2x-2-x\times 5-7x^{2}+7x=0$$

Combine 2x and 7x to get 9x.

$$9x-2-x\times 5-7x^{2}=0$$

Multiply -1 and 5 to get -5.

$$9x-2-5x-7x^{2}=0$$

Combine 9x and -5x to get 4x.

$$4x-2-7x^{2}=0$$

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.

$$-7x^{2}+4x-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 4 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-7\right)\left(-2\right)}}{2\left(-7\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-7\right)\left(-2\right)}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-4±\sqrt{16+28\left(-2\right)}}{2\left(-7\right)}$$

Multiply 28 times -2.

$$x=\frac{-4±\sqrt{16-56}}{2\left(-7\right)}$$

Add 16 to -56.

$$x=\frac{-4±\sqrt{-40}}{2\left(-7\right)}$$

Take the square root of -40.

$$x=\frac{-4±2\sqrt{10}i}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-4±2\sqrt{10}i}{-14}$$

Now solve the equation x=\frac{-4±2\sqrt{10}i}{-14} when ± is plus. Add -4 to 2i\sqrt{10}.

$$x=\frac{-4+2\sqrt{10}i}{-14}$$

Divide -4+2i\sqrt{10} by -14.

$$x=\frac{-\sqrt{10}i+2}{7}$$

Now solve the equation x=\frac{-4±2\sqrt{10}i}{-14} when ± is minus. Subtract 2i\sqrt{10} from -4.

$$x=\frac{-2\sqrt{10}i-4}{-14}$$

Divide -4-2i\sqrt{10} by -14.

$$x=\frac{2+\sqrt{10}i}{7}$$

The equation is now solved.

$$x=\frac{-\sqrt{10}i+2}{7} x=\frac{2+\sqrt{10}i}{7}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,1 since division by zero is not defined. Multiply both sides of the equation by x\left(x-1\right), the least common multiple of x,x-1.

$$\left(x-1\right)\times 2-x\times 5=7x\left(x-1\right)$$

Use the distributive property to multiply x-1 by 2.

$$2x-2-x\times 5=7x\left(x-1\right)$$

Use the distributive property to multiply 7x by x-1.

$$2x-2-x\times 5=7x^{2}-7x$$

Subtract 7x^{2} from both sides.

$$2x-2-x\times 5-7x^{2}=-7x$$

Add 7x to both sides.

$$2x-2-x\times 5-7x^{2}+7x=0$$

Combine 2x and 7x to get 9x.

$$9x-2-x\times 5-7x^{2}=0$$

Add 2 to both sides. Anything plus zero gives itself.

$$9x-x\times 5-7x^{2}=2$$

Multiply -1 and 5 to get -5.

$$9x-5x-7x^{2}=2$$

Combine 9x and -5x to get 4x.

$$4x-7x^{2}=2$$

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.

$$-7x^{2}+4x=2$$

Divide both sides by -7.

$$\frac{-7x^{2}+4x}{-7}=\frac{2}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\frac{4}{-7}x=\frac{2}{-7}$$

Divide 4 by -7.

$$x^{2}-\frac{4}{7}x=\frac{2}{-7}$$

Divide 2 by -7.

$$x^{2}-\frac{4}{7}x=-\frac{2}{7}$$

Divide -\frac{4}{7}, the coefficient of the x term, by 2 to get -\frac{2}{7}. Then add the square of -\frac{2}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{4}{7}x+\left(-\frac{2}{7}\right)^{2}=-\frac{2}{7}+\left(-\frac{2}{7}\right)^{2}$$

Square -\frac{2}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{4}{7}x+\frac{4}{49}=-\frac{2}{7}+\frac{4}{49}$$

Add -\frac{2}{7} to \frac{4}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{4}{7}x+\frac{4}{49}=-\frac{10}{49}$$

Factor x^{2}-\frac{4}{7}x+\frac{4}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{2}{7}\right)^{2}=-\frac{10}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2}{7}\right)^{2}}=\sqrt{-\frac{10}{49}}$$

Simplify.

$$x-\frac{2}{7}=\frac{\sqrt{10}i}{7} x-\frac{2}{7}=-\frac{\sqrt{10}i}{7}$$

Add \frac{2}{7} to both sides of the equation.

$$x=\frac{2+\sqrt{10}i}{7} x=\frac{-\sqrt{10}i+2}{7}$$