Solve (1.215-x)*30000+30000=36790/x

$$( 1.215 - x ) \times 30000 + 30000 = \frac { 36790 } { x }$$

Solve for x

$$x=\frac{\sqrt{321}}{1200}+1.1075\approx 1.122430394$$

$$x=-\frac{\sqrt{321}}{1200}+1.1075\approx 1.092569606$$

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.

$$\left(1.215-x\right)\times 30000x+x\times 30000=36790$$

Use the distributive property to multiply 1.215-x by 30000.

$$\left(36450-30000x\right)x+x\times 30000=36790$$

Use the distributive property to multiply 36450-30000x by x.

$$36450x-30000x^{2}+x\times 30000=36790$$

Combine 36450x and x\times 30000 to get 66450x.

$$66450x-30000x^{2}=36790$$

Subtract 36790 from both sides.

$$66450x-30000x^{2}-36790=0$$

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.

$$-30000x^{2}+66450x-36790=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -30000 for a, 66450 for b, and -36790 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-66450±\sqrt{66450^{2}-4\left(-30000\right)\left(-36790\right)}}{2\left(-30000\right)}$$

Square 66450.

$$x=\frac{-66450±\sqrt{4415602500-4\left(-30000\right)\left(-36790\right)}}{2\left(-30000\right)}$$

Multiply -4 times -30000.

$$x=\frac{-66450±\sqrt{4415602500+120000\left(-36790\right)}}{2\left(-30000\right)}$$

Multiply 120000 times -36790.

$$x=\frac{-66450±\sqrt{4415602500-4414800000}}{2\left(-30000\right)}$$

Add 4415602500 to -4414800000.

$$x=\frac{-66450±\sqrt{802500}}{2\left(-30000\right)}$$

Take the square root of 802500.

$$x=\frac{-66450±50\sqrt{321}}{2\left(-30000\right)}$$

Multiply 2 times -30000.

$$x=\frac{-66450±50\sqrt{321}}{-60000}$$

Now solve the equation x=\frac{-66450±50\sqrt{321}}{-60000} when ± is plus. Add -66450 to 50\sqrt{321}.

$$x=\frac{50\sqrt{321}-66450}{-60000}$$

Divide -66450+50\sqrt{321} by -60000.

$$x=-\frac{\sqrt{321}}{1200}+\frac{443}{400}$$

Now solve the equation x=\frac{-66450±50\sqrt{321}}{-60000} when ± is minus. Subtract 50\sqrt{321} from -66450.

$$x=\frac{-50\sqrt{321}-66450}{-60000}$$

Divide -66450-50\sqrt{321} by -60000.

$$x=\frac{\sqrt{321}}{1200}+\frac{443}{400}$$

The equation is now solved.

$$x=-\frac{\sqrt{321}}{1200}+\frac{443}{400} x=\frac{\sqrt{321}}{1200}+\frac{443}{400}$$

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.

$$\left(1.215-x\right)\times 30000x+x\times 30000=36790$$

Use the distributive property to multiply 1.215-x by 30000.

$$\left(36450-30000x\right)x+x\times 30000=36790$$

Use the distributive property to multiply 36450-30000x by x.

$$36450x-30000x^{2}+x\times 30000=36790$$

Combine 36450x and x\times 30000 to get 66450x.

$$66450x-30000x^{2}=36790$$

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.

$$-30000x^{2}+66450x=36790$$

Divide both sides by -30000.

$$\frac{-30000x^{2}+66450x}{-30000}=\frac{36790}{-30000}$$

Dividing by -30000 undoes the multiplication by -30000.

$$x^{2}+\frac{66450}{-30000}x=\frac{36790}{-30000}$$

Reduce the fraction \frac{66450}{-30000} to lowest terms by extracting and canceling out 150.

$$x^{2}-\frac{443}{200}x=\frac{36790}{-30000}$$

Reduce the fraction \frac{36790}{-30000} to lowest terms by extracting and canceling out 10.

$$x^{2}-\frac{443}{200}x=-\frac{3679}{3000}$$

Divide -\frac{443}{200}, the coefficient of the x term, by 2 to get -\frac{443}{400}. Then add the square of -\frac{443}{400} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{443}{200}x+\left(-\frac{443}{400}\right)^{2}=-\frac{3679}{3000}+\left(-\frac{443}{400}\right)^{2}$$

Square -\frac{443}{400} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{443}{200}x+\frac{196249}{160000}=-\frac{3679}{3000}+\frac{196249}{160000}$$

Add -\frac{3679}{3000} to \frac{196249}{160000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{443}{200}x+\frac{196249}{160000}=\frac{107}{480000}$$

Factor x^{2}-\frac{443}{200}x+\frac{196249}{160000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{443}{400}\right)^{2}=\frac{107}{480000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{443}{400}\right)^{2}}=\sqrt{\frac{107}{480000}}$$

Simplify.

$$x-\frac{443}{400}=\frac{\sqrt{321}}{1200} x-\frac{443}{400}=-\frac{\sqrt{321}}{1200}$$

Add \frac{443}{400} to both sides of the equation.

$$x=\frac{\sqrt{321}}{1200}+\frac{443}{400} x=-\frac{\sqrt{321}}{1200}+\frac{443}{400}$$