Solve (215a)^2/215(1-a)=116

$$\frac { ( 215 a ) ^ { 2 } } { 215 ( 1 - a ) } = 116$$

Solve for a

$$a=\frac{4\sqrt{1769}-58}{215}\approx 0.512734543$$

$$a=\frac{-4\sqrt{1769}-58}{215}\approx -1.052269427$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by 215\left(-a+1\right).

$$\left(215a\right)^{2}=24940\left(-a+1\right)$$

Expand \left(215a\right)^{2}.

$$215^{2}a^{2}=24940\left(-a+1\right)$$

Calculate 215 to the power of 2 and get 46225.

$$46225a^{2}=24940\left(-a+1\right)$$

Use the distributive property to multiply 24940 by -a+1.

$$46225a^{2}=-24940a+24940$$

Add 24940a to both sides.

$$46225a^{2}+24940a=24940$$

Subtract 24940 from both sides.

$$46225a^{2}+24940a-24940=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 46225 for a, 24940 for b, and -24940 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-24940±\sqrt{24940^{2}-4\times 46225\left(-24940\right)}}{2\times 46225}$$

Square 24940.

$$a=\frac{-24940±\sqrt{622003600-4\times 46225\left(-24940\right)}}{2\times 46225}$$

Multiply -4 times 46225.

$$a=\frac{-24940±\sqrt{622003600-184900\left(-24940\right)}}{2\times 46225}$$

Multiply -184900 times -24940.

$$a=\frac{-24940±\sqrt{622003600+4611406000}}{2\times 46225}$$

Add 622003600 to 4611406000.

$$a=\frac{-24940±\sqrt{5233409600}}{2\times 46225}$$

Take the square root of 5233409600.

$$a=\frac{-24940±1720\sqrt{1769}}{2\times 46225}$$

Multiply 2 times 46225.

$$a=\frac{-24940±1720\sqrt{1769}}{92450}$$

Now solve the equation a=\frac{-24940±1720\sqrt{1769}}{92450} when ± is plus. Add -24940 to 1720\sqrt{1769}.

$$a=\frac{1720\sqrt{1769}-24940}{92450}$$

Divide -24940+1720\sqrt{1769} by 92450.

$$a=\frac{4\sqrt{1769}-58}{215}$$

Now solve the equation a=\frac{-24940±1720\sqrt{1769}}{92450} when ± is minus. Subtract 1720\sqrt{1769} from -24940.

$$a=\frac{-1720\sqrt{1769}-24940}{92450}$$

Divide -24940-1720\sqrt{1769} by 92450.

$$a=\frac{-4\sqrt{1769}-58}{215}$$

The equation is now solved.

$$a=\frac{4\sqrt{1769}-58}{215} a=\frac{-4\sqrt{1769}-58}{215}$$

Steps for Completing the Square

Variable a cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by 215\left(-a+1\right).

$$\left(215a\right)^{2}=24940\left(-a+1\right)$$

Expand \left(215a\right)^{2}.

$$215^{2}a^{2}=24940\left(-a+1\right)$$

Calculate 215 to the power of 2 and get 46225.

$$46225a^{2}=24940\left(-a+1\right)$$

Use the distributive property to multiply 24940 by -a+1.

$$46225a^{2}=-24940a+24940$$

Add 24940a to both sides.

$$46225a^{2}+24940a=24940$$

Divide both sides by 46225.

$$\frac{46225a^{2}+24940a}{46225}=\frac{24940}{46225}$$

Dividing by 46225 undoes the multiplication by 46225.

$$a^{2}+\frac{24940}{46225}a=\frac{24940}{46225}$$

Reduce the fraction \frac{24940}{46225} to lowest terms by extracting and canceling out 215.

$$a^{2}+\frac{116}{215}a=\frac{24940}{46225}$$

Reduce the fraction \frac{24940}{46225} to lowest terms by extracting and canceling out 215.

$$a^{2}+\frac{116}{215}a=\frac{116}{215}$$

Divide \frac{116}{215}, the coefficient of the x term, by 2 to get \frac{58}{215}. Then add the square of \frac{58}{215} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+\frac{116}{215}a+\left(\frac{58}{215}\right)^{2}=\frac{116}{215}+\left(\frac{58}{215}\right)^{2}$$

Square \frac{58}{215} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+\frac{116}{215}a+\frac{3364}{46225}=\frac{116}{215}+\frac{3364}{46225}$$

Add \frac{116}{215} to \frac{3364}{46225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}+\frac{116}{215}a+\frac{3364}{46225}=\frac{28304}{46225}$$

Factor a^{2}+\frac{116}{215}a+\frac{3364}{46225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+\frac{58}{215}\right)^{2}=\frac{28304}{46225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{58}{215}\right)^{2}}=\sqrt{\frac{28304}{46225}}$$

Simplify.

$$a+\frac{58}{215}=\frac{4\sqrt{1769}}{215} a+\frac{58}{215}=-\frac{4\sqrt{1769}}{215}$$

Subtract \frac{58}{215} from both sides of the equation.

$$a=\frac{4\sqrt{1769}-58}{215} a=\frac{-4\sqrt{1769}-58}{215}$$