Solve 70/x+12-70/x=2.5

$$\frac{ 70 }{ x+12 } - \frac{ 70 }{ x } =2.5$$

Solve for x (complex solution)

$$x=-6+i\times 10\sqrt{3}\approx -6+17.320508076i$$

$$x=-i\times 10\sqrt{3}-6\approx -6-17.320508076i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -12,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+12\right), the least common multiple of x+12,x.

$$x\times 70-\left(x+12\right)\times 70=2.5x\left(x+12\right)$$

Use the distributive property to multiply x+12 by 70.

$$x\times 70-\left(70x+840\right)=2.5x\left(x+12\right)$$

To find the opposite of 70x+840, find the opposite of each term.

$$x\times 70-70x-840=2.5x\left(x+12\right)$$

Combine x\times 70 and -70x to get 0.

$$-840=2.5x\left(x+12\right)$$

Use the distributive property to multiply 2.5x by x+12.

$$-840=2.5x^{2}+30x$$

Swap sides so that all variable terms are on the left hand side.

$$2.5x^{2}+30x=-840$$

Add 840 to both sides.

$$2.5x^{2}+30x+840=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2.5 for a, 30 for b, and 840 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\times 2.5\times 840}}{2\times 2.5}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\times 2.5\times 840}}{2\times 2.5}$$

Multiply -4 times 2.5.

$$x=\frac{-30±\sqrt{900-10\times 840}}{2\times 2.5}$$

Multiply -10 times 840.

$$x=\frac{-30±\sqrt{900-8400}}{2\times 2.5}$$

Add 900 to -8400.

$$x=\frac{-30±\sqrt{-7500}}{2\times 2.5}$$

Take the square root of -7500.

$$x=\frac{-30±50\sqrt{3}i}{2\times 2.5}$$

Multiply 2 times 2.5.

$$x=\frac{-30±50\sqrt{3}i}{5}$$

Now solve the equation x=\frac{-30±50\sqrt{3}i}{5} when ± is plus. Add -30 to 50i\sqrt{3}.

$$x=\frac{-30+50\sqrt{3}i}{5}$$

Divide -30+50i\sqrt{3} by 5.

$$x=-6+10\sqrt{3}i$$

Now solve the equation x=\frac{-30±50\sqrt{3}i}{5} when ± is minus. Subtract 50i\sqrt{3} from -30.

$$x=\frac{-50\sqrt{3}i-30}{5}$$

Divide -30-50i\sqrt{3} by 5.

$$x=-10\sqrt{3}i-6$$

The equation is now solved.

$$x=-6+10\sqrt{3}i x=-10\sqrt{3}i-6$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -12,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+12\right), the least common multiple of x+12,x.

$$x\times 70-\left(x+12\right)\times 70=2.5x\left(x+12\right)$$

Use the distributive property to multiply x+12 by 70.

$$x\times 70-\left(70x+840\right)=2.5x\left(x+12\right)$$

To find the opposite of 70x+840, find the opposite of each term.

$$x\times 70-70x-840=2.5x\left(x+12\right)$$

Combine x\times 70 and -70x to get 0.

$$-840=2.5x\left(x+12\right)$$

Use the distributive property to multiply 2.5x by x+12.

$$-840=2.5x^{2}+30x$$

Swap sides so that all variable terms are on the left hand side.

$$2.5x^{2}+30x=-840$$

Divide both sides of the equation by 2.5, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{2.5x^{2}+30x}{2.5}=-\frac{840}{2.5}$$

Dividing by 2.5 undoes the multiplication by 2.5.

$$x^{2}+\frac{30}{2.5}x=-\frac{840}{2.5}$$

Divide 30 by 2.5 by multiplying 30 by the reciprocal of 2.5.

$$x^{2}+12x=-\frac{840}{2.5}$$

Divide -840 by 2.5 by multiplying -840 by the reciprocal of 2.5.

$$x^{2}+12x=-336$$

Divide 12, the coefficient of the x term, by 2 to get 6. Then add the square of 6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+12x+6^{2}=-336+6^{2}$$

Square 6.

$$x^{2}+12x+36=-336+36$$

Add -336 to 36.

$$x^{2}+12x+36=-300$$

Factor x^{2}+12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+6\right)^{2}=-300$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{-300}$$

Simplify.

$$x+6=10\sqrt{3}i x+6=-10\sqrt{3}i$$

Subtract 6 from both sides of the equation.

$$x=-6+10\sqrt{3}i x=-10\sqrt{3}i-6$$