Solve 1480/x-148/x+70=3

$$\frac { 1480 } { x } - \frac { 148 } { x + 70 } = 3$$

Solve for x

$$x=\frac{\sqrt{625521}}{3}+187\approx 450.632951911$$

$$x=-\frac{\sqrt{625521}}{3}+187\approx -76.632951911$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -70,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+70\right), the least common multiple of x,x+70.

$$\left(x+70\right)\times 1480-x\times 148=3x\left(x+70\right)$$

Use the distributive property to multiply x+70 by 1480.

$$1480x+103600-x\times 148=3x\left(x+70\right)$$

Use the distributive property to multiply 3x by x+70.

$$1480x+103600-x\times 148=3x^{2}+210x$$

Subtract 3x^{2} from both sides.

$$1480x+103600-x\times 148-3x^{2}=210x$$

Subtract 210x from both sides.

$$1480x+103600-x\times 148-3x^{2}-210x=0$$

Combine 1480x and -210x to get 1270x.

$$1270x+103600-x\times 148-3x^{2}=0$$

Multiply -1 and 148 to get -148.

$$1270x+103600-148x-3x^{2}=0$$

Combine 1270x and -148x to get 1122x.

$$1122x+103600-3x^{2}=0$$

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.

$$-3x^{2}+1122x+103600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 1122 for b, and 103600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1122±\sqrt{1122^{2}-4\left(-3\right)\times 103600}}{2\left(-3\right)}$$

Square 1122.

$$x=\frac{-1122±\sqrt{1258884-4\left(-3\right)\times 103600}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-1122±\sqrt{1258884+12\times 103600}}{2\left(-3\right)}$$

Multiply 12 times 103600.

$$x=\frac{-1122±\sqrt{1258884+1243200}}{2\left(-3\right)}$$

Add 1258884 to 1243200.

$$x=\frac{-1122±\sqrt{2502084}}{2\left(-3\right)}$$

Take the square root of 2502084.

$$x=\frac{-1122±2\sqrt{625521}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-1122±2\sqrt{625521}}{-6}$$

Now solve the equation x=\frac{-1122±2\sqrt{625521}}{-6} when ± is plus. Add -1122 to 2\sqrt{625521}.

$$x=\frac{2\sqrt{625521}-1122}{-6}$$

Divide -1122+2\sqrt{625521} by -6.

$$x=-\frac{\sqrt{625521}}{3}+187$$

Now solve the equation x=\frac{-1122±2\sqrt{625521}}{-6} when ± is minus. Subtract 2\sqrt{625521} from -1122.

$$x=\frac{-2\sqrt{625521}-1122}{-6}$$

Divide -1122-2\sqrt{625521} by -6.

$$x=\frac{\sqrt{625521}}{3}+187$$

The equation is now solved.

$$x=-\frac{\sqrt{625521}}{3}+187 x=\frac{\sqrt{625521}}{3}+187$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -70,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+70\right), the least common multiple of x,x+70.

$$\left(x+70\right)\times 1480-x\times 148=3x\left(x+70\right)$$

Use the distributive property to multiply x+70 by 1480.

$$1480x+103600-x\times 148=3x\left(x+70\right)$$

Use the distributive property to multiply 3x by x+70.

$$1480x+103600-x\times 148=3x^{2}+210x$$

Subtract 3x^{2} from both sides.

$$1480x+103600-x\times 148-3x^{2}=210x$$

Subtract 210x from both sides.

$$1480x+103600-x\times 148-3x^{2}-210x=0$$

Combine 1480x and -210x to get 1270x.

$$1270x+103600-x\times 148-3x^{2}=0$$

Subtract 103600 from both sides. Anything subtracted from zero gives its negation.

$$1270x-x\times 148-3x^{2}=-103600$$

Multiply -1 and 148 to get -148.

$$1270x-148x-3x^{2}=-103600$$

Combine 1270x and -148x to get 1122x.

$$1122x-3x^{2}=-103600$$

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.

$$-3x^{2}+1122x=-103600$$

Divide both sides by -3.

$$\frac{-3x^{2}+1122x}{-3}=-\frac{103600}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{1122}{-3}x=-\frac{103600}{-3}$$

Divide 1122 by -3.

$$x^{2}-374x=-\frac{103600}{-3}$$

Divide -103600 by -3.

$$x^{2}-374x=\frac{103600}{3}$$

Divide -374, the coefficient of the x term, by 2 to get -187. Then add the square of -187 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-374x+\left(-187\right)^{2}=\frac{103600}{3}+\left(-187\right)^{2}$$

Square -187.

$$x^{2}-374x+34969=\frac{103600}{3}+34969$$

Add \frac{103600}{3} to 34969.

$$x^{2}-374x+34969=\frac{208507}{3}$$

Factor x^{2}-374x+34969. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-187\right)^{2}=\frac{208507}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-187\right)^{2}}=\sqrt{\frac{208507}{3}}$$

Simplify.

$$x-187=\frac{\sqrt{625521}}{3} x-187=-\frac{\sqrt{625521}}{3}$$

Add 187 to both sides of the equation.

$$x=\frac{\sqrt{625521}}{3}+187 x=-\frac{\sqrt{625521}}{3}+187$$