Solve 55x=720/x+600

$$55x = \frac{ 720 }{ x } +600$$

Solve for x

$$x = -\frac{12}{11} = -1\frac{1}{11} \approx -1.090909091$$

$$x=12$$

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.

$$55xx=720+x\times 600$$

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

$$55x^{2}=720+x\times 600$$

Subtract 720 from both sides.

$$55x^{2}-720=x\times 600$$

Subtract x\times 600 from both sides.

$$55x^{2}-720-x\times 600=0$$

Multiply -1 and 600 to get -600.

$$55x^{2}-720-600x=0$$

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.

$$55x^{2}-600x-720=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 55 for a, -600 for b, and -720 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-600\right)±\sqrt{\left(-600\right)^{2}-4\times 55\left(-720\right)}}{2\times 55}$$

Square -600.

$$x=\frac{-\left(-600\right)±\sqrt{360000-4\times 55\left(-720\right)}}{2\times 55}$$

Multiply -4 times 55.

$$x=\frac{-\left(-600\right)±\sqrt{360000-220\left(-720\right)}}{2\times 55}$$

Multiply -220 times -720.

$$x=\frac{-\left(-600\right)±\sqrt{360000+158400}}{2\times 55}$$

Add 360000 to 158400.

$$x=\frac{-\left(-600\right)±\sqrt{518400}}{2\times 55}$$

Take the square root of 518400.

$$x=\frac{-\left(-600\right)±720}{2\times 55}$$

The opposite of -600 is 600.

$$x=\frac{600±720}{2\times 55}$$

Multiply 2 times 55.

$$x=\frac{600±720}{110}$$

Now solve the equation x=\frac{600±720}{110} when ± is plus. Add 600 to 720.

$$x=\frac{1320}{110}$$

Divide 1320 by 110.

$$x=12$$

Now solve the equation x=\frac{600±720}{110} when ± is minus. Subtract 720 from 600.

$$x=-\frac{120}{110}$$

Reduce the fraction \frac{-120}{110} to lowest terms by extracting and canceling out 10.

$$x=-\frac{12}{11}$$

The equation is now solved.

$$x=12 x=-\frac{12}{11}$$

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.

$$55xx=720+x\times 600$$

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

$$55x^{2}=720+x\times 600$$

Subtract x\times 600 from both sides.

$$55x^{2}-x\times 600=720$$

Multiply -1 and 600 to get -600.

$$55x^{2}-600x=720$$

Divide both sides by 55.

$$\frac{55x^{2}-600x}{55}=\frac{720}{55}$$

Dividing by 55 undoes the multiplication by 55.

$$x^{2}+\left(-\frac{600}{55}\right)x=\frac{720}{55}$$

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

$$x^{2}-\frac{120}{11}x=\frac{720}{55}$$

Reduce the fraction \frac{720}{55} to lowest terms by extracting and canceling out 5.

$$x^{2}-\frac{120}{11}x=\frac{144}{11}$$

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.

$$x^{2}-\frac{120}{11}x+\left(-\frac{60}{11}\right)^{2}=\frac{144}{11}+\left(-\frac{60}{11}\right)^{2}$$

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

$$x^{2}-\frac{120}{11}x+\frac{3600}{121}=\frac{144}{11}+\frac{3600}{121}$$

Add \frac{144}{11} to \frac{3600}{121} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{120}{11}x+\frac{3600}{121}=\frac{5184}{121}$$

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}.

$$\left(x-\frac{60}{11}\right)^{2}=\frac{5184}{121}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{60}{11}\right)^{2}}=\sqrt{\frac{5184}{121}}$$

Simplify.

$$x-\frac{60}{11}=\frac{72}{11} x-\frac{60}{11}=-\frac{72}{11}$$

Add \frac{60}{11} to both sides of the equation.

$$x=12 x=-\frac{12}{11}$$