Solve 1700left(25-yright)1800y=44000

$$1700 \left( 25-y \right) 1800y = 44000$$

Solve for y

$$y=\frac{\sqrt{40636885}}{510}+\frac{25}{2}\approx 24.999424823$$

$$y=-\frac{\sqrt{40636885}}{510}+\frac{25}{2}\approx 0.000575177$$

Steps Using the Quadratic Formula

Multiply 1700 and 1800 to get 3060000.

$$3060000\left(25-y\right)y=44000$$

Use the distributive property to multiply 3060000 by 25-y.

$$\left(76500000-3060000y\right)y=44000$$

Use the distributive property to multiply 76500000-3060000y by y.

$$76500000y-3060000y^{2}=44000$$

Subtract 44000 from both sides.

$$76500000y-3060000y^{2}-44000=0$$

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.

$$-3060000y^{2}[phone]y-44000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3060000 for a, 76500000 for b, and -44000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-76500000±\sqrt{76500000^{2}-4\left(-3060000\right)\left(-44000\right)}}{2\left(-3060000\right)}$$

Square 76500000.

$$y=\frac{-76500000±\sqrt{5852250000000000-4\left(-3060000\right)\left(-44000\right)}}{2\left(-3060000\right)}$$

Multiply -4 times -3060000.

$$y=\frac{-76500000±\sqrt{5852250000000000+12240000\left(-44000\right)}}{2\left(-3060000\right)}$$

Multiply 12240000 times -44000.

$$y=\frac{-76500000±\sqrt{5852250000000000-538560000000}}{2\left(-3060000\right)}$$

Add 5852250000000000 to -538560000000.

$$y=\frac{-76500000±\sqrt{5851711440000000}}{2\left(-3060000\right)}$$

Take the square root of 5851711440000000.

$$y=\frac{-76500000±12000\sqrt{40636885}}{2\left(-3060000\right)}$$

Multiply 2 times -3060000.

$$y=\frac{-76500000±12000\sqrt{40636885}}{-6120000}$$

Now solve the equation y=\frac{-76500000±12000\sqrt{40636885}}{-6120000} when ± is plus. Add -76500000 to 12000\sqrt{40636885}.

$$y=\frac{12000\sqrt{40636885}-76500000}{-6120000}$$

Divide -76500000+12000\sqrt{40636885} by -6120000.

$$y=-\frac{\sqrt{40636885}}{510}+\frac{25}{2}$$

Now solve the equation y=\frac{-76500000±12000\sqrt{40636885}}{-6120000} when ± is minus. Subtract 12000\sqrt{40636885} from -76500000.

$$y=\frac{-12000\sqrt{40636885}-76500000}{-6120000}$$

Divide -76500000-12000\sqrt{40636885} by -6120000.

$$y=\frac{\sqrt{40636885}}{510}+\frac{25}{2}$$

The equation is now solved.

$$y=-\frac{\sqrt{40636885}}{510}+\frac{25}{2} y=\frac{\sqrt{40636885}}{510}+\frac{25}{2}$$

Steps for Completing the Square

Multiply 1700 and 1800 to get 3060000.

$$3060000\left(25-y\right)y=44000$$

Use the distributive property to multiply 3060000 by 25-y.

$$\left(76500000-3060000y\right)y=44000$$

Use the distributive property to multiply 76500000-3060000y by y.

$$76500000y-3060000y^{2}=44000$$

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.

$$-3060000y^{2}[phone]y=44000$$

Divide both sides by -3060000.

$$\frac{-3060000y^{2}[phone]y}{-3060000}=\frac{44000}{-3060000}$$

Dividing by -3060000 undoes the multiplication by -3060000.

$$y^{2}+\frac{76500000}{-3060000}y=\frac{44000}{-3060000}$$

Divide 76500000 by -3060000.

$$y^{2}-25y=\frac{44000}{-3060000}$$

Reduce the fraction \frac{44000}{-3060000} to lowest terms by extracting and canceling out 4000.

$$y^{2}-25y=-\frac{11}{765}$$

Divide -25, the coefficient of the x term, by 2 to get -\frac{25}{2}. Then add the square of -\frac{25}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-25y+\left(-\frac{25}{2}\right)^{2}=-\frac{11}{765}+\left(-\frac{25}{2}\right)^{2}$$

Square -\frac{25}{2} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-25y+\frac{625}{4}=-\frac{11}{765}+\frac{625}{4}$$

Add -\frac{11}{765} to \frac{625}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-25y+\frac{625}{4}=\frac{478081}{3060}$$

Factor y^{2}-25y+\frac{625}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{25}{2}\right)^{2}=\frac{478081}{3060}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{25}{2}\right)^{2}}=\sqrt{\frac{478081}{3060}}$$

Simplify.

$$y-\frac{25}{2}=\frac{\sqrt{40636885}}{510} y-\frac{25}{2}=-\frac{\sqrt{40636885}}{510}$$

Add \frac{25}{2} to both sides of the equation.

$$y=\frac{\sqrt{40636885}}{510}+\frac{25}{2} y=-\frac{\sqrt{40636885}}{510}+\frac{25}{2}$$