Solve 3*1600/x=6000/x+2quad3x

$$3 \cdot \frac { 1600 } { x } = \frac { 6000 } { x + 2 } \quad 3 x$$

Solve for x

$$x=\frac{2\sqrt{31}+2}{15}\approx 0.875701915$$

$$x=\frac{2-2\sqrt{31}}{15}\approx -0.609035248$$

Steps Using the Quadratic Formula

Cancel out 3 on both sides.

$$\frac{1600}{x}=\frac{6000}{x+2}x$$

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+2\right), the least common multiple of x,x+2.

$$\left(x+2\right)\times 1600=x\times 6000x$$

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

$$\left(x+2\right)\times 1600=x^{2}\times 6000$$

Use the distributive property to multiply x+2 by 1600.

$$1600x+3200=x^{2}\times 6000$$

Subtract x^{2}\times 6000 from both sides.

$$1600x+3200-x^{2}\times 6000=0$$

Multiply -1 and 6000 to get -6000.

$$1600x+3200-6000x^{2}=0$$

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.

$$-6000x^{2}+1600x+3200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6000 for a, 1600 for b, and 3200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1600±\sqrt{1600^{2}-4\left(-6000\right)\times 3200}}{2\left(-6000\right)}$$

Square 1600.

$$x=\frac{-1600±\sqrt{2560000-4\left(-6000\right)\times 3200}}{2\left(-6000\right)}$$

Multiply -4 times -6000.

$$x=\frac{-1600±\sqrt{2560000+24000\times 3200}}{2\left(-6000\right)}$$

Multiply 24000 times 3200.

$$x=\frac{-1600±\sqrt{2560000+76800000}}{2\left(-6000\right)}$$

Add 2560000 to 76800000.

$$x=\frac{-1600±\sqrt{79360000}}{2\left(-6000\right)}$$

Take the square root of 79360000.

$$x=\frac{-1600±1600\sqrt{31}}{2\left(-6000\right)}$$

Multiply 2 times -6000.

$$x=\frac{-1600±1600\sqrt{31}}{-12000}$$

Now solve the equation x=\frac{-1600±1600\sqrt{31}}{-12000} when ± is plus. Add -1600 to 1600\sqrt{31}.

$$x=\frac{1600\sqrt{31}-1600}{-12000}$$

Divide -1600+1600\sqrt{31} by -12000.

$$x=\frac{2-2\sqrt{31}}{15}$$

Now solve the equation x=\frac{-1600±1600\sqrt{31}}{-12000} when ± is minus. Subtract 1600\sqrt{31} from -1600.

$$x=\frac{-1600\sqrt{31}-1600}{-12000}$$

Divide -1600-1600\sqrt{31} by -12000.

$$x=\frac{2\sqrt{31}+2}{15}$$

The equation is now solved.

$$x=\frac{2-2\sqrt{31}}{15} x=\frac{2\sqrt{31}+2}{15}$$

Steps for Completing the Square

Cancel out 3 on both sides.

$$\frac{1600}{x}=\frac{6000}{x+2}x$$

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+2\right), the least common multiple of x,x+2.

$$\left(x+2\right)\times 1600=x\times 6000x$$

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

$$\left(x+2\right)\times 1600=x^{2}\times 6000$$

Use the distributive property to multiply x+2 by 1600.

$$1600x+3200=x^{2}\times 6000$$

Subtract x^{2}\times 6000 from both sides.

$$1600x+3200-x^{2}\times 6000=0$$

Multiply -1 and 6000 to get -6000.

$$1600x+3200-6000x^{2}=0$$

Subtract 3200 from both sides. Anything subtracted from zero gives its negation.

$$1600x-6000x^{2}=-3200$$

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.

$$-6000x^{2}+1600x=-3200$$

Divide both sides by -6000.

$$\frac{-6000x^{2}+1600x}{-6000}=-\frac{3200}{-6000}$$

Dividing by -6000 undoes the multiplication by -6000.

$$x^{2}+\frac{1600}{-6000}x=-\frac{3200}{-6000}$$

Reduce the fraction \frac{1600}{-6000} to lowest terms by extracting and canceling out 400.

$$x^{2}-\frac{4}{15}x=-\frac{3200}{-6000}$$

Reduce the fraction \frac{-3200}{-6000} to lowest terms by extracting and canceling out 400.

$$x^{2}-\frac{4}{15}x=\frac{8}{15}$$

Divide -\frac{4}{15}, the coefficient of the x term, by 2 to get -\frac{2}{15}. Then add the square of -\frac{2}{15} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{4}{15}x+\left(-\frac{2}{15}\right)^{2}=\frac{8}{15}+\left(-\frac{2}{15}\right)^{2}$$

Square -\frac{2}{15} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{4}{15}x+\frac{4}{225}=\frac{8}{15}+\frac{4}{225}$$

Add \frac{8}{15} to \frac{4}{225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{4}{15}x+\frac{4}{225}=\frac{124}{225}$$

Factor x^{2}-\frac{4}{15}x+\frac{4}{225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{2}{15}\right)^{2}=\frac{124}{225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2}{15}\right)^{2}}=\sqrt{\frac{124}{225}}$$

Simplify.

$$x-\frac{2}{15}=\frac{2\sqrt{31}}{15} x-\frac{2}{15}=-\frac{2\sqrt{31}}{15}$$

Add \frac{2}{15} to both sides of the equation.

$$x=\frac{2\sqrt{31}+2}{15} x=\frac{2-2\sqrt{31}}{15}$$