Solve (8.314*273)^2*(a^2-0.1a)=2.31*10^6

$$( 8.314 \times 273 ) ^ { 2 } \times ( a ^ { 2 } - 0.1 a ) = 2.31 \times 10 ^ { 6 }$$

Solve for a

$$a=\frac{\sqrt{232287909489321}}{22697220}+0.05\approx 0.721491679$$

$$a=-\frac{\sqrt{232287909489321}}{22697220}+0.05\approx -0.621491679$$

Steps Using the Quadratic Formula

Multiply 8.314 and 273 to get 2269.722.

$$2269.722^{2}\left(a^{2}-0.1a\right)=2.31\times 10^{6}$$

Calculate 2269.722 to the power of 2 and get 5151637.957284.

$$5151637.957284\left(a^{2}-0.1a\right)=2.31\times 10^{6}$$

Use the distributive property to multiply 5151637.957284 by a^{2}-0.1a.

$$5151637.957284a^{2}-515163.7957284a=2.31\times 10^{6}$$

Calculate 10 to the power of 6 and get 1000000.

$$5151637.957284a^{2}-515163.7957284a=2.31\times 1000000$$

Multiply 2.31 and 1000000 to get 2310000.

$$5151637.957284a^{2}-515163.7957284a=2310000$$

Subtract 2310000 from both sides.

$$5151637.957284a^{2}-515163.7957284a-2310000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5151637.957284 for a, -515163.7957284 for b, and -2310000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-515163.7957284\right)±\sqrt{\left(-515163.7957284\right)^{2}-4\times 5151637.957284\left(-2310000\right)}}{2\times 5151637.957284}$$

Square -515163.7957284 by squaring both the numerator and the denominator of the fraction.

$$a=\frac{-\left(-515163.7957284\right)±\sqrt{265393736429.29264208656656-4\times 5151637.957284\left(-2310000\right)}}{2\times 5151637.957284}$$

Multiply -4 times 5151637.957284.

$$a=\frac{-\left(-515163.7957284\right)±\sqrt{265393736429.29264208656656-20606551.829136\left(-2310000\right)}}{2\times 5151637.957284}$$

Multiply -20606551.829136 times -2310000.

$$a=\frac{-\left(-515163.7957284\right)±\sqrt{265393736429.29264208656656+47601134725304.16}}{2\times 5151637.957284}$$

Add 265393736429.29264208656656 to 47601134725304.16 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a=\frac{-\left(-515163.7957284\right)±\sqrt{47866528461733.45264208656656}}{2\times 5151637.957284}$$

Take the square root of 47866528461733.45264208656656.

$$a=\frac{-\left(-515163.7957284\right)±\frac{1134861\sqrt{232287909489321}}{2500000}}{2\times 5151637.957284}$$

The opposite of -515163.7957284 is 515163.7957284.

$$a=\frac{515163.7957284±\frac{1134861\sqrt{232287909489321}}{2500000}}{2\times 5151637.957284}$$

Multiply 2 times 5151637.957284.

$$a=\frac{515163.7957284±\frac{1134861\sqrt{232287909489321}}{2500000}}{10303275.914568}$$

Now solve the equation a=\frac{515163.7957284±\frac{1134861\sqrt{232287909489321}}{2500000}}{10303275.914568} when ± is plus. Add 515163.7957284 to \frac{1134861\sqrt{232287909489321}}{2500000}.

$$a=\frac{1134861\sqrt{232287909489321}[phone]}{2500000\times 10303275.914568}$$

Divide \frac{1287909489321+1134861\sqrt{232287909489321}}{2500000} by 10303275.914568 by multiplying \frac{1287909489321+1134861\sqrt{232287909489321}}{2500000} by the reciprocal of 10303275.914568.

$$a=\frac{\sqrt{232287909489321}}{22697220}+\frac{1}{20}$$

Now solve the equation a=\frac{515163.7957284±\frac{1134861\sqrt{232287909489321}}{2500000}}{10303275.914568} when ± is minus. Subtract \frac{1134861\sqrt{232287909489321}}{2500000} from 515163.7957284.

$$a=\frac{1287909489321-1134861\sqrt{232287909489321}}{2500000\times 10303275.914568}$$

Divide \frac{1287909489321-1134861\sqrt{232287909489321}}{2500000} by 10303275.914568 by multiplying \frac{1287909489321-1134861\sqrt{232287909489321}}{2500000} by the reciprocal of 10303275.914568.

$$a=-\frac{\sqrt{232287909489321}}{22697220}+\frac{1}{20}$$

The equation is now solved.

$$a=\frac{\sqrt{232287909489321}}{22697220}+\frac{1}{20} a=-\frac{\sqrt{232287909489321}}{22697220}+\frac{1}{20}$$

Steps for Completing the Square

Multiply 8.314 and 273 to get 2269.722.

$$2269.722^{2}\left(a^{2}-0.1a\right)=2.31\times 10^{6}$$

Calculate 2269.722 to the power of 2 and get 5151637.957284.

$$5151637.957284\left(a^{2}-0.1a\right)=2.31\times 10^{6}$$

Use the distributive property to multiply 5151637.957284 by a^{2}-0.1a.

$$5151637.957284a^{2}-515163.7957284a=2.31\times 10^{6}$$

Calculate 10 to the power of 6 and get 1000000.

$$5151637.957284a^{2}-515163.7957284a=2.31\times 1000000$$

Multiply 2.31 and 1000000 to get 2310000.

$$5151637.957284a^{2}-515163.7957284a=2310000$$

Divide both sides of the equation by 5151637.957284, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{5151637.957284a^{2}-515163.7957284a}{5151637.957284}=\frac{2310000}{5151637.957284}$$

Dividing by 5151637.957284 undoes the multiplication by 5151637.957284.

$$a^{2}+\left(-\frac{515163.7957284}{5151637.957284}\right)a=\frac{2310000}{5151637.957284}$$

Divide -515163.7957284 by 5151637.957284 by multiplying -515163.7957284 by the reciprocal of 5151637.957284.

$$a^{2}-0.1a=\frac{2310000}{5151637.957284}$$

Divide 2310000 by 5151637.957284 by multiplying 2310000 by the reciprocal of 5151637.957284.

$$a^{2}-0.1a=\frac{27500000000}{61329023301}$$

Divide -0.1, the coefficient of the x term, by 2 to get -0.05. Then add the square of -0.05 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-0.1a+\left(-0.05\right)^{2}=\frac{27500000000}{61329023301}+\left(-0.05\right)^{2}$$

Square -0.05 by squaring both the numerator and the denominator of the fraction.

$$a^{2}-0.1a+0.0025=\frac{27500000000}{61329023301}+0.0025$$

Add \frac{27500000000}{61329023301} to 0.0025 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}-0.1a+0.0025=\frac{11061329023301}{24531609320400}$$

Factor a^{2}-0.1a+0.0025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-0.05\right)^{2}=\frac{11061329023301}{24531609320400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-0.05\right)^{2}}=\sqrt{\frac{11061329023301}{24531609320400}}$$

Simplify.

$$a-0.05=\frac{\sqrt{232287909489321}}{22697220} a-0.05=-\frac{\sqrt{232287909489321}}{22697220}$$

Add 0.05 to both sides of the equation.

$$a=\frac{\sqrt{232287909489321}}{22697220}+\frac{1}{20} a=-\frac{\sqrt{232287909489321}}{22697220}+\frac{1}{20}$$