Solve 1500/a*1000+1200/2a+240/a*1000=150000

$$\frac { 1500 } { a } \times 1000 + \frac { \frac { 1200 } { 2 a + 240 } } { a } \times 1000 = 150000$$

Solve for a (complex solution)

$$a=\sqrt{4229}-55\approx 10.030761952$$

$$a=-\left(\sqrt{4229}+55\right)\approx -120.030761952$$

Steps Using the Quadratic Formula

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

$$1500\times 1000+\frac{1200}{2a+240}\times 1000=150000a$$

Multiply 1500 and 1000 to get 1500000.

$$1500000+\frac{1200}{2a+240}\times 1000=150000a$$

Express \frac{1200}{2a+240}\times 1000 as a single fraction.

$$1500000+\frac{1200\times 1000}{2a+240}=150000a$$

Factor 2a+240.

$$1500000+\frac{1200\times 1000}{2\left(a+120\right)}=150000a$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1500000 times \frac{2\left(a+120\right)}{2\left(a+120\right)}.

$$\frac{1500000\times 2\left(a+120\right)}{2\left(a+120\right)}+\frac{1200\times 1000}{2\left(a+120\right)}=150000a$$

Since \frac{1500000\times 2\left(a+120\right)}{2\left(a+120\right)} and \frac{1200\times 1000}{2\left(a+120\right)} have the same denominator, add them by adding their numerators.

$$\frac{1500000\times 2\left(a+120\right)+1200\times 1000}{2\left(a+120\right)}=150000a$$

Do the multiplications in 1500000\times 2\left(a+120\right)+1200\times 1000.

$$\frac{3000000a+360000000+1200000}{2\left(a+120\right)}=150000a$$

Combine like terms in 3000000a+360000000+1200000.

$$\frac{3000000a+361200000}{2\left(a+120\right)}=150000a$$

Factor the expressions that are not already factored in \frac{3000000a+361200000}{2\left(a+120\right)}.

$$\frac{600000\left(5a+602\right)}{2\left(a+120\right)}=150000a$$

Cancel out 2 in both numerator and denominator.

$$\frac{300000\left(5a+602\right)}{a+120}=150000a$$

Use the distributive property to multiply 300000 by 5a+602.

$$\frac{1500000a+180600000}{a+120}=150000a$$

Subtract 150000a from both sides.

$$\frac{1500000a+180600000}{a+120}-150000a=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -150000a times \frac{a+120}{a+120}.

$$\frac{1500000a+180600000}{a+120}+\frac{-150000a\left(a+120\right)}{a+120}=0$$

Since \frac{1500000a+180600000}{a+120} and \frac{-150000a\left(a+120\right)}{a+120} have the same denominator, add them by adding their numerators.

$$\frac{1500000a+180600000-150000a\left(a+120\right)}{a+120}=0$$

Do the multiplications in 1500000a+180600000-150000a\left(a+120\right).

$$\frac{1500000a+180600000-150000a^{2}-18000000a}{a+120}=0$$

Combine like terms in 1500000a+180600000-150000a^{2}-18000000a.

$$\frac{-16500000a+180600000-150000a^{2}}{a+120}=0$$

Variable a cannot be equal to -120 since division by zero is not defined. Multiply both sides of the equation by a+120.

$$-16500000a+180600000-150000a^{2}=0$$

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.

$$-150000a^{2}-16500000a+180600000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -150000 for a, -16500000 for b, and 180600000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-16500000\right)±\sqrt{\left(-16500000\right)^{2}-4\left(-150000\right)\times 180600000}}{2\left(-150000\right)}$$

Square -16500000.

$$a=\frac{-\left(-16500000\right)±\sqrt{272250000000000-4\left(-150000\right)\times 180600000}}{2\left(-150000\right)}$$

Multiply -4 times -150000.

$$a=\frac{-\left(-16500000\right)±\sqrt{272250000000000+600000\times 180600000}}{2\left(-150000\right)}$$

Multiply 600000 times 180600000.

$$a=\frac{-\left(-16500000\right)±\sqrt{272250000000000+108360000000000}}{2\left(-150000\right)}$$

Add 272250000000000 to 108360000000000.

$$a=\frac{-\left(-16500000\right)±\sqrt{380610000000000}}{2\left(-150000\right)}$$

Take the square root of 380610000000000.

$$a=\frac{-\left(-16500000\right)±300000\sqrt{4229}}{2\left(-150000\right)}$$

The opposite of -16500000 is 16500000.

$$a=\frac{16500000±300000\sqrt{4229}}{2\left(-150000\right)}$$

Multiply 2 times -150000.

$$a=\frac{16500000±300000\sqrt{4229}}{-300000}$$

Now solve the equation a=\frac{16500000±300000\sqrt{4229}}{-300000} when ± is plus. Add 16500000 to 300000\sqrt{4229}.

$$a=\frac{300000\sqrt{4229}[phone]}{-300000}$$

Divide 16500000+300000\sqrt{4229} by -300000.

$$a=-\left(\sqrt{4229}+55\right)$$

Now solve the equation a=\frac{16500000±300000\sqrt{4229}}{-300000} when ± is minus. Subtract 300000\sqrt{4229} from 16500000.

$$a=\frac{16500000-300000\sqrt{4229}}{-300000}$$

Divide 16500000-300000\sqrt{4229} by -300000.

$$a=\sqrt{4229}-55$$

The equation is now solved.

$$a=-\left(\sqrt{4229}+55\right) a=\sqrt{4229}-55$$

Steps for Completing the Square

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

$$1500\times 1000+\frac{1200}{2a+240}\times 1000=150000a$$

Multiply 1500 and 1000 to get 1500000.

$$1500000+\frac{1200}{2a+240}\times 1000=150000a$$

Express \frac{1200}{2a+240}\times 1000 as a single fraction.

$$1500000+\frac{1200\times 1000}{2a+240}=150000a$$

Factor 2a+240.

$$1500000+\frac{1200\times 1000}{2\left(a+120\right)}=150000a$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1500000 times \frac{2\left(a+120\right)}{2\left(a+120\right)}.

$$\frac{1500000\times 2\left(a+120\right)}{2\left(a+120\right)}+\frac{1200\times 1000}{2\left(a+120\right)}=150000a$$

Since \frac{1500000\times 2\left(a+120\right)}{2\left(a+120\right)} and \frac{1200\times 1000}{2\left(a+120\right)} have the same denominator, add them by adding their numerators.

$$\frac{1500000\times 2\left(a+120\right)+1200\times 1000}{2\left(a+120\right)}=150000a$$

Do the multiplications in 1500000\times 2\left(a+120\right)+1200\times 1000.

$$\frac{3000000a+360000000+1200000}{2\left(a+120\right)}=150000a$$

Combine like terms in 3000000a+360000000+1200000.

$$\frac{3000000a+361200000}{2\left(a+120\right)}=150000a$$

Factor the expressions that are not already factored in \frac{3000000a+361200000}{2\left(a+120\right)}.

$$\frac{600000\left(5a+602\right)}{2\left(a+120\right)}=150000a$$

Cancel out 2 in both numerator and denominator.

$$\frac{300000\left(5a+602\right)}{a+120}=150000a$$

Use the distributive property to multiply 300000 by 5a+602.

$$\frac{1500000a+180600000}{a+120}=150000a$$

Subtract 150000a from both sides.

$$\frac{1500000a+180600000}{a+120}-150000a=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -150000a times \frac{a+120}{a+120}.

$$\frac{1500000a+180600000}{a+120}+\frac{-150000a\left(a+120\right)}{a+120}=0$$

Since \frac{1500000a+180600000}{a+120} and \frac{-150000a\left(a+120\right)}{a+120} have the same denominator, add them by adding their numerators.

$$\frac{1500000a+180600000-150000a\left(a+120\right)}{a+120}=0$$

Do the multiplications in 1500000a+180600000-150000a\left(a+120\right).

$$\frac{1500000a+180600000-150000a^{2}-18000000a}{a+120}=0$$

Combine like terms in 1500000a+180600000-150000a^{2}-18000000a.

$$\frac{-16500000a+180600000-150000a^{2}}{a+120}=0$$

Variable a cannot be equal to -120 since division by zero is not defined. Multiply both sides of the equation by a+120.

$$-16500000a+180600000-150000a^{2}=0$$

Subtract 180600000 from both sides. Anything subtracted from zero gives its negation.

$$-16500000a-150000a^{2}=-180600000$$

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.

$$-150000a^{2}-16500000a=-180600000$$

Divide both sides by -150000.

$$\frac{-150000a^{2}-16500000a}{-150000}=-\frac{180600000}{-150000}$$

Dividing by -150000 undoes the multiplication by -150000.

$$a^{2}+\left(-\frac{16500000}{-150000}\right)a=-\frac{180600000}{-150000}$$

Divide -16500000 by -150000.

$$a^{2}+110a=-\frac{180600000}{-150000}$$

Divide -180600000 by -150000.

$$a^{2}+110a=1204$$

Divide 110, the coefficient of the x term, by 2 to get 55. Then add the square of 55 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+110a+55^{2}=1204+55^{2}$$

Square 55.

$$a^{2}+110a+3025=1204+3025$$

Add 1204 to 3025.

$$a^{2}+110a+3025=4229$$

Factor a^{2}+110a+3025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+55\right)^{2}=4229$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+55\right)^{2}}=\sqrt{4229}$$

Simplify.

$$a+55=\sqrt{4229} a+55=-\sqrt{4229}$$

Subtract 55 from both sides of the equation.

$$a=\sqrt{4229}-55 a=-\sqrt{4229}-55$$

Solve for a

$$a=\sqrt{4229}-55\approx 10.030761952$$

$$a=-\sqrt{4229}-55\approx -120.030761952$$

Steps Using the Quadratic Formula

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

$$1500\times 1000+\frac{1200}{2a+240}\times 1000=150000a$$

Multiply 1500 and 1000 to get 1500000.

$$1500000+\frac{1200}{2a+240}\times 1000=150000a$$

Express \frac{1200}{2a+240}\times 1000 as a single fraction.

$$1500000+\frac{1200\times 1000}{2a+240}=150000a$$

Factor 2a+240.

$$1500000+\frac{1200\times 1000}{2\left(a+120\right)}=150000a$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1500000 times \frac{2\left(a+120\right)}{2\left(a+120\right)}.

$$\frac{1500000\times 2\left(a+120\right)}{2\left(a+120\right)}+\frac{1200\times 1000}{2\left(a+120\right)}=150000a$$

Since \frac{1500000\times 2\left(a+120\right)}{2\left(a+120\right)} and \frac{1200\times 1000}{2\left(a+120\right)} have the same denominator, add them by adding their numerators.

$$\frac{1500000\times 2\left(a+120\right)+1200\times 1000}{2\left(a+120\right)}=150000a$$

Do the multiplications in 1500000\times 2\left(a+120\right)+1200\times 1000.

$$\frac{3000000a+360000000+1200000}{2\left(a+120\right)}=150000a$$

Combine like terms in 3000000a+360000000+1200000.

$$\frac{3000000a+361200000}{2\left(a+120\right)}=150000a$$

Factor the expressions that are not already factored in \frac{3000000a+361200000}{2\left(a+120\right)}.

$$\frac{600000\left(5a+602\right)}{2\left(a+120\right)}=150000a$$

Cancel out 2 in both numerator and denominator.

$$\frac{300000\left(5a+602\right)}{a+120}=150000a$$

Use the distributive property to multiply 300000 by 5a+602.

$$\frac{1500000a+180600000}{a+120}=150000a$$

Subtract 150000a from both sides.

$$\frac{1500000a+180600000}{a+120}-150000a=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -150000a times \frac{a+120}{a+120}.

$$\frac{1500000a+180600000}{a+120}+\frac{-150000a\left(a+120\right)}{a+120}=0$$

Since \frac{1500000a+180600000}{a+120} and \frac{-150000a\left(a+120\right)}{a+120} have the same denominator, add them by adding their numerators.

$$\frac{1500000a+180600000-150000a\left(a+120\right)}{a+120}=0$$

Do the multiplications in 1500000a+180600000-150000a\left(a+120\right).

$$\frac{1500000a+180600000-150000a^{2}-18000000a}{a+120}=0$$

Combine like terms in 1500000a+180600000-150000a^{2}-18000000a.

$$\frac{-16500000a+180600000-150000a^{2}}{a+120}=0$$

Variable a cannot be equal to -120 since division by zero is not defined. Multiply both sides of the equation by a+120.

$$-16500000a+180600000-150000a^{2}=0$$

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.

$$-150000a^{2}-16500000a+180600000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -150000 for a, -16500000 for b, and 180600000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-16500000\right)±\sqrt{\left(-16500000\right)^{2}-4\left(-150000\right)\times 180600000}}{2\left(-150000\right)}$$

Square -16500000.

$$a=\frac{-\left(-16500000\right)±\sqrt{272250000000000-4\left(-150000\right)\times 180600000}}{2\left(-150000\right)}$$

Multiply -4 times -150000.

$$a=\frac{-\left(-16500000\right)±\sqrt{272250000000000+600000\times 180600000}}{2\left(-150000\right)}$$

Multiply 600000 times 180600000.

$$a=\frac{-\left(-16500000\right)±\sqrt{272250000000000+108360000000000}}{2\left(-150000\right)}$$

Add 272250000000000 to 108360000000000.

$$a=\frac{-\left(-16500000\right)±\sqrt{380610000000000}}{2\left(-150000\right)}$$

Take the square root of 380610000000000.

$$a=\frac{-\left(-16500000\right)±300000\sqrt{4229}}{2\left(-150000\right)}$$

The opposite of -16500000 is 16500000.

$$a=\frac{16500000±300000\sqrt{4229}}{2\left(-150000\right)}$$

Multiply 2 times -150000.

$$a=\frac{16500000±300000\sqrt{4229}}{-300000}$$

Now solve the equation a=\frac{16500000±300000\sqrt{4229}}{-300000} when ± is plus. Add 16500000 to 300000\sqrt{4229}.

$$a=\frac{300000\sqrt{4229}[phone]}{-300000}$$

Divide 16500000+300000\sqrt{4229} by -300000.

$$a=-\left(\sqrt{4229}+55\right)$$

Now solve the equation a=\frac{16500000±300000\sqrt{4229}}{-300000} when ± is minus. Subtract 300000\sqrt{4229} from 16500000.

$$a=\frac{16500000-300000\sqrt{4229}}{-300000}$$

Divide 16500000-300000\sqrt{4229} by -300000.

$$a=\sqrt{4229}-55$$

The equation is now solved.

$$a=-\left(\sqrt{4229}+55\right) a=\sqrt{4229}-55$$

Steps for Completing the Square

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

$$1500\times 1000+\frac{1200}{2a+240}\times 1000=150000a$$

Multiply 1500 and 1000 to get 1500000.

$$1500000+\frac{1200}{2a+240}\times 1000=150000a$$

Express \frac{1200}{2a+240}\times 1000 as a single fraction.

$$1500000+\frac{1200\times 1000}{2a+240}=150000a$$

Factor 2a+240.

$$1500000+\frac{1200\times 1000}{2\left(a+120\right)}=150000a$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1500000 times \frac{2\left(a+120\right)}{2\left(a+120\right)}.

$$\frac{1500000\times 2\left(a+120\right)}{2\left(a+120\right)}+\frac{1200\times 1000}{2\left(a+120\right)}=150000a$$

Since \frac{1500000\times 2\left(a+120\right)}{2\left(a+120\right)} and \frac{1200\times 1000}{2\left(a+120\right)} have the same denominator, add them by adding their numerators.

$$\frac{1500000\times 2\left(a+120\right)+1200\times 1000}{2\left(a+120\right)}=150000a$$

Do the multiplications in 1500000\times 2\left(a+120\right)+1200\times 1000.

$$\frac{3000000a+360000000+1200000}{2\left(a+120\right)}=150000a$$

Combine like terms in 3000000a+360000000+1200000.

$$\frac{3000000a+361200000}{2\left(a+120\right)}=150000a$$

Factor the expressions that are not already factored in \frac{3000000a+361200000}{2\left(a+120\right)}.

$$\frac{600000\left(5a+602\right)}{2\left(a+120\right)}=150000a$$

Cancel out 2 in both numerator and denominator.

$$\frac{300000\left(5a+602\right)}{a+120}=150000a$$

Use the distributive property to multiply 300000 by 5a+602.

$$\frac{1500000a+180600000}{a+120}=150000a$$

Subtract 150000a from both sides.

$$\frac{1500000a+180600000}{a+120}-150000a=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -150000a times \frac{a+120}{a+120}.

$$\frac{1500000a+180600000}{a+120}+\frac{-150000a\left(a+120\right)}{a+120}=0$$

Since \frac{1500000a+180600000}{a+120} and \frac{-150000a\left(a+120\right)}{a+120} have the same denominator, add them by adding their numerators.

$$\frac{1500000a+180600000-150000a\left(a+120\right)}{a+120}=0$$

Do the multiplications in 1500000a+180600000-150000a\left(a+120\right).

$$\frac{1500000a+180600000-150000a^{2}-18000000a}{a+120}=0$$

Combine like terms in 1500000a+180600000-150000a^{2}-18000000a.

$$\frac{-16500000a+180600000-150000a^{2}}{a+120}=0$$

Variable a cannot be equal to -120 since division by zero is not defined. Multiply both sides of the equation by a+120.

$$-16500000a+180600000-150000a^{2}=0$$

Subtract 180600000 from both sides. Anything subtracted from zero gives its negation.

$$-16500000a-150000a^{2}=-180600000$$

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.

$$-150000a^{2}-16500000a=-180600000$$

Divide both sides by -150000.

$$\frac{-150000a^{2}-16500000a}{-150000}=-\frac{180600000}{-150000}$$

Dividing by -150000 undoes the multiplication by -150000.

$$a^{2}+\left(-\frac{16500000}{-150000}\right)a=-\frac{180600000}{-150000}$$

Divide -16500000 by -150000.

$$a^{2}+110a=-\frac{180600000}{-150000}$$

Divide -180600000 by -150000.

$$a^{2}+110a=1204$$

Divide 110, the coefficient of the x term, by 2 to get 55. Then add the square of 55 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+110a+55^{2}=1204+55^{2}$$

Square 55.

$$a^{2}+110a+3025=1204+3025$$

Add 1204 to 3025.

$$a^{2}+110a+3025=4229$$

Factor a^{2}+110a+3025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+55\right)^{2}=4229$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+55\right)^{2}}=\sqrt{4229}$$

Simplify.

$$a+55=\sqrt{4229} a+55=-\sqrt{4229}$$

Subtract 55 from both sides of the equation.

$$a=\sqrt{4229}-55 a=-\sqrt{4229}-55$$