Solve 1/2a*(60/a)^2=2500

$$\frac { 1 } { 2 } a \cdot ( \frac { 60 } { a } ) ^ { 2 } = 2500$$

Solve for a

$$a=\frac{18}{25}=0.72$$

Steps Using Factoring

To raise \frac{60}{a} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{1}{2}a\times \frac{60^{2}}{a^{2}}=2500$$

Multiply \frac{1}{2} times \frac{60^{2}}{a^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{60^{2}}{2a^{2}}a=2500$$

Calculate 60 to the power of 2 and get 3600.

$$\frac{3600}{2a^{2}}a=2500$$

Express \frac{3600}{2a^{2}}a as a single fraction.

$$\frac{3600a}{2a^{2}}=2500$$

Cancel out 2 in both numerator and denominator.

$$\frac{1800a}{a^{2}}=2500$$

Subtract 2500 from both sides.

$$\frac{1800a}{a^{2}}-2500=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2500 times \frac{a^{2}}{a^{2}}.

$$\frac{1800a}{a^{2}}-\frac{2500a^{2}}{a^{2}}=0$$

Since \frac{1800a}{a^{2}} and \frac{2500a^{2}}{a^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1800a-2500a^{2}}{a^{2}}=0$$

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by a^{2}.

$$1800a-2500a^{2}=0$$

Factor out a.

$$a\left(1800-2500a\right)=0$$

To find equation solutions, solve a=0 and 1800-2500a=0.

$$a=0 a=\frac{18}{25}$$

Variable a cannot be equal to 0.

$$a=\frac{18}{25}$$

Steps Using the Quadratic Formula

To raise \frac{60}{a} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{1}{2}a\times \frac{60^{2}}{a^{2}}=2500$$

Multiply \frac{1}{2} times \frac{60^{2}}{a^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{60^{2}}{2a^{2}}a=2500$$

Calculate 60 to the power of 2 and get 3600.

$$\frac{3600}{2a^{2}}a=2500$$

Express \frac{3600}{2a^{2}}a as a single fraction.

$$\frac{3600a}{2a^{2}}=2500$$

Cancel out 2 in both numerator and denominator.

$$\frac{1800a}{a^{2}}=2500$$

Subtract 2500 from both sides.

$$\frac{1800a}{a^{2}}-2500=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2500 times \frac{a^{2}}{a^{2}}.

$$\frac{1800a}{a^{2}}-\frac{2500a^{2}}{a^{2}}=0$$

Since \frac{1800a}{a^{2}} and \frac{2500a^{2}}{a^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1800a-2500a^{2}}{a^{2}}=0$$

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by a^{2}.

$$1800a-2500a^{2}=0$$

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.

$$-2500a^{2}+1800a=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2500 for a, 1800 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-1800±\sqrt{1800^{2}}}{2\left(-2500\right)}$$

Take the square root of 1800^{2}.

$$a=\frac{-1800±1800}{2\left(-2500\right)}$$

Multiply 2 times -2500.

$$a=\frac{-1800±1800}{-5000}$$

Now solve the equation a=\frac{-1800±1800}{-5000} when ± is plus. Add -1800 to 1800.

$$a=\frac{0}{-5000}$$

Divide 0 by -5000.

$$a=0$$

Now solve the equation a=\frac{-1800±1800}{-5000} when ± is minus. Subtract 1800 from -1800.

$$a=-\frac{3600}{-5000}$$

Reduce the fraction \frac{-3600}{-5000} to lowest terms by extracting and canceling out 200.

$$a=\frac{18}{25}$$

The equation is now solved.

$$a=0 a=\frac{18}{25}$$

Variable a cannot be equal to 0.

$$a=\frac{18}{25}$$

Steps for Completing the Square

To raise \frac{60}{a} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{1}{2}a\times \frac{60^{2}}{a^{2}}=2500$$

Multiply \frac{1}{2} times \frac{60^{2}}{a^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{60^{2}}{2a^{2}}a=2500$$

Calculate 60 to the power of 2 and get 3600.

$$\frac{3600}{2a^{2}}a=2500$$

Express \frac{3600}{2a^{2}}a as a single fraction.

$$\frac{3600a}{2a^{2}}=2500$$

Cancel out 2 in both numerator and denominator.

$$\frac{1800a}{a^{2}}=2500$$

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by a^{2}.

$$1800a=2500a^{2}$$

Subtract 2500a^{2} from both sides.

$$1800a-2500a^{2}=0$$

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.

$$-2500a^{2}+1800a=0$$

Divide both sides by -2500.

$$\frac{-2500a^{2}+1800a}{-2500}=\frac{0}{-2500}$$

Dividing by -2500 undoes the multiplication by -2500.

$$a^{2}+\frac{1800}{-2500}a=\frac{0}{-2500}$$

Reduce the fraction \frac{1800}{-2500} to lowest terms by extracting and canceling out 100.

$$a^{2}-\frac{18}{25}a=\frac{0}{-2500}$$

Divide 0 by -2500.

$$a^{2}-\frac{18}{25}a=0$$

Divide -\frac{18}{25}, the coefficient of the x term, by 2 to get -\frac{9}{25}. Then add the square of -\frac{9}{25} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{18}{25}a+\left(-\frac{9}{25}\right)^{2}=\left(-\frac{9}{25}\right)^{2}$$

Square -\frac{9}{25} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{18}{25}a+\frac{81}{625}=\frac{81}{625}$$

Factor a^{2}-\frac{18}{25}a+\frac{81}{625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{9}{25}\right)^{2}=\frac{81}{625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{9}{25}\right)^{2}}=\sqrt{\frac{81}{625}}$$

Simplify.

$$a-\frac{9}{25}=\frac{9}{25} a-\frac{9}{25}=-\frac{9}{25}$$

Add \frac{9}{25} to both sides of the equation.

$$a=\frac{18}{25} a=0$$

Variable a cannot be equal to 0.

$$a=\frac{18}{25}$$