Solve 405=30/15+y+30/25y

$$405 = \frac { 30 } { 15 + y } + \frac { 30 } { 25 y }$$

Solve for y

$$y=\frac{\sqrt{101546329}-10073}{1350}\approx 0.002977665$$

$$y=\frac{-\sqrt{101546329}-10073}{1350}\approx -14.925940628$$

Steps Using the Quadratic Formula

Variable y cannot be equal to any of the values -15,0 since division by zero is not defined. Multiply both sides of the equation by 25y\left(y+15\right), the least common multiple of 15+y,25y.

$$10125y\left(y+15\right)=25y\times 30+\left(y+15\right)\times 30$$

Use the distributive property to multiply 10125y by y+15.

$$10125y^{2}+151875y=25y\times 30+\left(y+15\right)\times 30$$

Multiply 25 and 30 to get 750.

$$10125y^{2}+151875y=750y+\left(y+15\right)\times 30$$

Use the distributive property to multiply y+15 by 30.

$$10125y^{2}+151875y=750y+30y+450$$

Combine 750y and 30y to get 780y.

$$10125y^{2}+151875y=780y+450$$

Subtract 780y from both sides.

$$10125y^{2}+151875y-780y=450$$

Combine 151875y and -780y to get 151095y.

$$10125y^{2}+151095y=450$$

Subtract 450 from both sides.

$$10125y^{2}+151095y-450=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10125 for a, 151095 for b, and -450 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-151095±\sqrt{151095^{2}-4\times 10125\left(-450\right)}}{2\times 10125}$$

Square 151095.

$$y=\frac{-151095±\sqrt{22829699025-4\times 10125\left(-450\right)}}{2\times 10125}$$

Multiply -4 times 10125.

$$y=\frac{-151095±\sqrt{22829699025-40500\left(-450\right)}}{2\times 10125}$$

Multiply -40500 times -450.

$$y=\frac{-151095±\sqrt{22829699025+18225000}}{2\times 10125}$$

Add 22829699025 to 18225000.

$$y=\frac{-151095±\sqrt{22847924025}}{2\times 10125}$$

Take the square root of 22847924025.

$$y=\frac{-151095±15\sqrt{101546329}}{2\times 10125}$$

Multiply 2 times 10125.

$$y=\frac{-151095±15\sqrt{101546329}}{20250}$$

Now solve the equation y=\frac{-151095±15\sqrt{101546329}}{20250} when ± is plus. Add -151095 to 15\sqrt{101546329}.

$$y=\frac{15\sqrt{101546329}-151095}{20250}$$

Divide -151095+15\sqrt{101546329} by 20250.

$$y=\frac{\sqrt{101546329}-10073}{1350}$$

Now solve the equation y=\frac{-151095±15\sqrt{101546329}}{20250} when ± is minus. Subtract 15\sqrt{101546329} from -151095.

$$y=\frac{-15\sqrt{101546329}-151095}{20250}$$

Divide -151095-15\sqrt{101546329} by 20250.

$$y=\frac{-\sqrt{101546329}-10073}{1350}$$

The equation is now solved.

$$y=\frac{\sqrt{101546329}-10073}{1350} y=\frac{-\sqrt{101546329}-10073}{1350}$$

Steps for Completing the Square

Variable y cannot be equal to any of the values -15,0 since division by zero is not defined. Multiply both sides of the equation by 25y\left(y+15\right), the least common multiple of 15+y,25y.

$$10125y\left(y+15\right)=25y\times 30+\left(y+15\right)\times 30$$

Use the distributive property to multiply 10125y by y+15.

$$10125y^{2}+151875y=25y\times 30+\left(y+15\right)\times 30$$

Multiply 25 and 30 to get 750.

$$10125y^{2}+151875y=750y+\left(y+15\right)\times 30$$

Use the distributive property to multiply y+15 by 30.

$$10125y^{2}+151875y=750y+30y+450$$

Combine 750y and 30y to get 780y.

$$10125y^{2}+151875y=780y+450$$

Subtract 780y from both sides.

$$10125y^{2}+151875y-780y=450$$

Combine 151875y and -780y to get 151095y.

$$10125y^{2}+151095y=450$$

Divide both sides by 10125.

$$\frac{10125y^{2}+151095y}{10125}=\frac{450}{10125}$$

Dividing by 10125 undoes the multiplication by 10125.

$$y^{2}+\frac{151095}{10125}y=\frac{450}{10125}$$

Reduce the fraction \frac{151095}{10125} to lowest terms by extracting and canceling out 15.

$$y^{2}+\frac{10073}{675}y=\frac{450}{10125}$$

Reduce the fraction \frac{450}{10125} to lowest terms by extracting and canceling out 225.

$$y^{2}+\frac{10073}{675}y=\frac{2}{45}$$

Divide \frac{10073}{675}, the coefficient of the x term, by 2 to get \frac{10073}{1350}. Then add the square of \frac{10073}{1350} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{10073}{675}y+\left(\frac{10073}{1350}\right)^{2}=\frac{2}{45}+\left(\frac{10073}{1350}\right)^{2}$$

Square \frac{10073}{1350} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{10073}{675}y+\frac{101465329}{1822500}=\frac{2}{45}+\frac{101465329}{1822500}$$

Add \frac{2}{45} to \frac{101465329}{1822500} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}+\frac{10073}{675}y+\frac{101465329}{1822500}=\frac{101546329}{1822500}$$

Factor y^{2}+\frac{10073}{675}y+\frac{101465329}{1822500}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{10073}{1350}\right)^{2}=\frac{101546329}{1822500}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{10073}{1350}\right)^{2}}=\sqrt{\frac{101546329}{1822500}}$$

Simplify.

$$y+\frac{10073}{1350}=\frac{\sqrt{101546329}}{1350} y+\frac{10073}{1350}=-\frac{\sqrt{101546329}}{1350}$$

Subtract \frac{10073}{1350} from both sides of the equation.

$$y=\frac{\sqrt{101546329}-10073}{1350} y=\frac{-\sqrt{101546329}-10073}{1350}$$