Solve 55x=700/x-600

$$55 x = \frac { 700 } { x } - 600$$

Solve for x

$$x = \frac{2 \sqrt{1285} - 60}{11} \approx 1.06307212$$

$$x=\frac{-2\sqrt{1285}-60}{11}\approx -11.972163029$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$55xx=700+x\left(-600\right)$$

Multiply x and x to get x^{2}.

$$55x^{2}=700+x\left(-600\right)$$

Subtract 700 from both sides.

$$55x^{2}-700=x\left(-600\right)$$

Subtract x\left(-600\right) from both sides.

$$55x^{2}-700-x\left(-600\right)=0$$

Multiply -1 and -600 to get 600.

$$55x^{2}-700+600x=0$$

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.

$$55x^{2}+600x-700=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 55 for a, 600 for b, and -700 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-600±\sqrt{600^{2}-4\times 55\left(-700\right)}}{2\times 55}$$

Square 600.

$$x=\frac{-600±\sqrt{360000-4\times 55\left(-700\right)}}{2\times 55}$$

Multiply -4 times 55.

$$x=\frac{-600±\sqrt{360000-220\left(-700\right)}}{2\times 55}$$

Multiply -220 times -700.

$$x=\frac{-600±\sqrt{360000+154000}}{2\times 55}$$

Add 360000 to 154000.

$$x=\frac{-600±\sqrt{514000}}{2\times 55}$$

Take the square root of 514000.

$$x=\frac{-600±20\sqrt{1285}}{2\times 55}$$

Multiply 2 times 55.

$$x=\frac{-600±20\sqrt{1285}}{110}$$

Now solve the equation x=\frac{-600±20\sqrt{1285}}{110} when ± is plus. Add -600 to 20\sqrt{1285}.

$$x=\frac{20\sqrt{1285}-600}{110}$$

Divide -600+20\sqrt{1285} by 110.

$$x=\frac{2\sqrt{1285}-60}{11}$$

Now solve the equation x=\frac{-600±20\sqrt{1285}}{110} when ± is minus. Subtract 20\sqrt{1285} from -600.

$$x=\frac{-20\sqrt{1285}-600}{110}$$

Divide -600-20\sqrt{1285} by 110.

$$x=\frac{-2\sqrt{1285}-60}{11}$$

The equation is now solved.

$$x=\frac{2\sqrt{1285}-60}{11} x=\frac{-2\sqrt{1285}-60}{11}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$55xx=700+x\left(-600\right)$$

Multiply x and x to get x^{2}.

$$55x^{2}=700+x\left(-600\right)$$

Subtract x\left(-600\right) from both sides.

$$55x^{2}-x\left(-600\right)=700$$

Multiply -1 and -600 to get 600.

$$55x^{2}+600x=700$$

Divide both sides by 55.

$$\frac{55x^{2}+600x}{55}=\frac{700}{55}$$

Dividing by 55 undoes the multiplication by 55.

$$x^{2}+\frac{600}{55}x=\frac{700}{55}$$

Reduce the fraction \frac{600}{55} to lowest terms by extracting and canceling out 5.

$$x^{2}+\frac{120}{11}x=\frac{700}{55}$$

Reduce the fraction \frac{700}{55} to lowest terms by extracting and canceling out 5.

$$x^{2}+\frac{120}{11}x=\frac{140}{11}$$

Divide \frac{120}{11}, the coefficient of the x term, by 2 to get \frac{60}{11}. Then add the square of \frac{60}{11} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{120}{11}x+\left(\frac{60}{11}\right)^{2}=\frac{140}{11}+\left(\frac{60}{11}\right)^{2}$$

Square \frac{60}{11} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{120}{11}x+\frac{3600}{121}=\frac{140}{11}+\frac{3600}{121}$$

Add \frac{140}{11} to \frac{3600}{121} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{120}{11}x+\frac{3600}{121}=\frac{5140}{121}$$

Factor x^{2}+\frac{120}{11}x+\frac{3600}{121}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{60}{11}\right)^{2}=\frac{5140}{121}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{60}{11}\right)^{2}}=\sqrt{\frac{5140}{121}}$$

Simplify.

$$x+\frac{60}{11}=\frac{2\sqrt{1285}}{11} x+\frac{60}{11}=-\frac{2\sqrt{1285}}{11}$$

Subtract \frac{60}{11} from both sides of the equation.

$$x=\frac{2\sqrt{1285}-60}{11} x=\frac{-2\sqrt{1285}-60}{11}$$