Solve 201/1+frac{1{2}}=a_1^2/1-frac{1{4}}

$$\frac { 201 } { 1 + \frac { 1 } { 2 } } = \frac { a _ { 1 } ^ { 2 } } { 1 - \frac { 1 } { 4 } }$$

Solve for a_1

$$a_{1} = \frac{\sqrt{403}}{2} \approx 10.03742995$$

$$a_{1} = -\frac{\sqrt{403}}{2} \approx -10.03742995$$

Steps by Finding Square Root

Add 1 and \frac{1}{2} to get \frac{3}{2}.

$$\frac{201}{\frac{3}{2}}=\frac{a_{1}^{2}\times 1-\frac{1}{4}}{1-\frac{1}{4}}$$

Divide 201 by \frac{3}{2} by multiplying 201 by the reciprocal of \frac{3}{2}.

$$201\times \frac{2}{3}=\frac{a_{1}^{2}\times 1-\frac{1}{4}}{1-\frac{1}{4}}$$

Multiply 201 and \frac{2}{3} to get 134.

$$134=\frac{a_{1}^{2}\times 1-\frac{1}{4}}{1-\frac{1}{4}}$$

Subtract \frac{1}{4} from 1 to get \frac{3}{4}.

$$134=\frac{a_{1}^{2}\times 1-\frac{1}{4}}{\frac{3}{4}}$$

Divide each term of a_{1}^{2}\times 1-\frac{1}{4} by \frac{3}{4} to get \frac{a_{1}^{2}\times 1}{\frac{3}{4}}+\frac{-\frac{1}{4}}{\frac{3}{4}}.

$$134=\frac{a_{1}^{2}\times 1}{\frac{3}{4}}+\frac{-\frac{1}{4}}{\frac{3}{4}}$$

Divide a_{1}^{2}\times 1 by \frac{3}{4} to get a_{1}^{2}\times \frac{4}{3}.

$$134=a_{1}^{2}\times \frac{4}{3}+\frac{-\frac{1}{4}}{\frac{3}{4}}$$

Divide -\frac{1}{4} by \frac{3}{4} by multiplying -\frac{1}{4} by the reciprocal of \frac{3}{4}.

$$134=a_{1}^{2}\times \frac{4}{3}-\frac{1}{4}\times \frac{4}{3}$$

Multiply -\frac{1}{4} and \frac{4}{3} to get -\frac{1}{3}.

$$134=a_{1}^{2}\times \frac{4}{3}-\frac{1}{3}$$

Swap sides so that all variable terms are on the left hand side.

$$a_{1}^{2}\times \frac{4}{3}-\frac{1}{3}=134$$

Add \frac{1}{3} to both sides.

$$a_{1}^{2}\times \frac{4}{3}=134+\frac{1}{3}$$

Add 134 and \frac{1}{3} to get \frac{403}{3}.

$$a_{1}^{2}\times \frac{4}{3}=\frac{403}{3}$$

Multiply both sides by \frac{3}{4}, the reciprocal of \frac{4}{3}.

$$a_{1}^{2}=\frac{403}{3}\times \frac{3}{4}$$

Multiply \frac{403}{3} and \frac{3}{4} to get \frac{403}{4}.

$$a_{1}^{2}=\frac{403}{4}$$

Take the square root of both sides of the equation.

$$a_{1}=\frac{\sqrt{403}}{2} a_{1}=-\frac{\sqrt{403}}{2}$$

Steps Using the Quadratic Formula

Add 1 and \frac{1}{2} to get \frac{3}{2}.

$$\frac{201}{\frac{3}{2}}=\frac{a_{1}^{2}\times 1-\frac{1}{4}}{1-\frac{1}{4}}$$

Divide 201 by \frac{3}{2} by multiplying 201 by the reciprocal of \frac{3}{2}.

$$201\times \frac{2}{3}=\frac{a_{1}^{2}\times 1-\frac{1}{4}}{1-\frac{1}{4}}$$

Multiply 201 and \frac{2}{3} to get 134.

$$134=\frac{a_{1}^{2}\times 1-\frac{1}{4}}{1-\frac{1}{4}}$$

Subtract \frac{1}{4} from 1 to get \frac{3}{4}.

$$134=\frac{a_{1}^{2}\times 1-\frac{1}{4}}{\frac{3}{4}}$$

Divide each term of a_{1}^{2}\times 1-\frac{1}{4} by \frac{3}{4} to get \frac{a_{1}^{2}\times 1}{\frac{3}{4}}+\frac{-\frac{1}{4}}{\frac{3}{4}}.

$$134=\frac{a_{1}^{2}\times 1}{\frac{3}{4}}+\frac{-\frac{1}{4}}{\frac{3}{4}}$$

Divide a_{1}^{2}\times 1 by \frac{3}{4} to get a_{1}^{2}\times \frac{4}{3}.

$$134=a_{1}^{2}\times \frac{4}{3}+\frac{-\frac{1}{4}}{\frac{3}{4}}$$

Divide -\frac{1}{4} by \frac{3}{4} by multiplying -\frac{1}{4} by the reciprocal of \frac{3}{4}.

$$134=a_{1}^{2}\times \frac{4}{3}-\frac{1}{4}\times \frac{4}{3}$$

Multiply -\frac{1}{4} and \frac{4}{3} to get -\frac{1}{3}.

$$134=a_{1}^{2}\times \frac{4}{3}-\frac{1}{3}$$

Swap sides so that all variable terms are on the left hand side.

$$a_{1}^{2}\times \frac{4}{3}-\frac{1}{3}=134$$

Subtract 134 from both sides.

$$a_{1}^{2}\times \frac{4}{3}-\frac{1}{3}-134=0$$

Subtract 134 from -\frac{1}{3} to get -\frac{403}{3}.

$$a_{1}^{2}\times \frac{4}{3}-\frac{403}{3}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$\frac{4}{3}a_{1}^{2}-\frac{403}{3}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{4}{3} for a, 0 for b, and -\frac{403}{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a_{1}=\frac{0±\sqrt{0^{2}-4\times \frac{4}{3}\left(-\frac{403}{3}\right)}}{2\times \frac{4}{3}}$$

Square 0.

$$a_{1}=\frac{0±\sqrt{-4\times \frac{4}{3}\left(-\frac{403}{3}\right)}}{2\times \frac{4}{3}}$$

Multiply -4 times \frac{4}{3}.

$$a_{1}=\frac{0±\sqrt{-\frac{16}{3}\left(-\frac{403}{3}\right)}}{2\times \frac{4}{3}}$$

Multiply -\frac{16}{3} times -\frac{403}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$a_{1}=\frac{0±\sqrt{\frac{6448}{9}}}{2\times \frac{4}{3}}$$

Take the square root of \frac{6448}{9}.

$$a_{1}=\frac{0±\frac{4\sqrt{403}}{3}}{2\times \frac{4}{3}}$$

Multiply 2 times \frac{4}{3}.

$$a_{1}=\frac{0±\frac{4\sqrt{403}}{3}}{\frac{8}{3}}$$

Now solve the equation a_{1}=\frac{0±\frac{4\sqrt{403}}{3}}{\frac{8}{3}} when ± is plus.

$$a_{1}=\frac{\sqrt{403}}{2}$$

Now solve the equation a_{1}=\frac{0±\frac{4\sqrt{403}}{3}}{\frac{8}{3}} when ± is minus.

$$a_{1}=-\frac{\sqrt{403}}{2}$$

The equation is now solved.

$$a_{1}=\frac{\sqrt{403}}{2} a_{1}=-\frac{\sqrt{403}}{2}$$