Solve (a-2)^2+(8/9a-16/9)^2+(3-3a/2)^2=34^2

$$( a - 2 ) ^ { 2 } + ( \frac { 8 } { 9 } a - \frac { 16 } { 9 } ) ^ { 2 } + ( 3 - \frac { 3 a } { 2 } ) ^ { 2 } = 34 ^ { 2 }$$

Solve for a

$$a=\frac{36\sqrt{1309}}{77}+2\approx 18.91537378$$

$$a=-\frac{36\sqrt{1309}}{77}+2\approx -14.91537378$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(a-2\right)^{2}.

$$a^{2}-4a+4+\left(\frac{8}{9}a-\frac{16}{9}\right)^{2}+\left(3-\frac{3a}{2}\right)^{2}=34^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{8}{9}a-\frac{16}{9}\right)^{2}.

$$a^{2}-4a+4+\frac{64}{81}a^{2}-\frac{256}{81}a+\frac{256}{81}+\left(3-\frac{3a}{2}\right)^{2}=34^{2}$$

Combine a^{2} and \frac{64}{81}a^{2} to get \frac{145}{81}a^{2}.

$$\frac{145}{81}a^{2}-4a+4-\frac{256}{81}a+\frac{256}{81}+\left(3-\frac{3a}{2}\right)^{2}=34^{2}$$

Combine -4a and -\frac{256}{81}a to get -\frac{580}{81}a.

$$\frac{145}{81}a^{2}-\frac{580}{81}a+4+\frac{256}{81}+\left(3-\frac{3a}{2}\right)^{2}=34^{2}$$

Add 4 and \frac{256}{81} to get \frac{580}{81}.

$$\frac{145}{81}a^{2}-\frac{580}{81}a+\frac{580}{81}+\left(3-\frac{3a}{2}\right)^{2}=34^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3-\frac{3a}{2}\right)^{2}.

$$\frac{145}{81}a^{2}-\frac{580}{81}a+\frac{580}{81}+9+6\left(-\frac{3a}{2}\right)+\left(-\frac{3a}{2}\right)^{2}=34^{2}$$

Cancel out 2, the greatest common factor in 6 and 2.

$$\frac{145}{81}a^{2}-\frac{580}{81}a+\frac{580}{81}+9-3\times 3a+\left(-\frac{3a}{2}\right)^{2}=34^{2}$$

Calculate -\frac{3a}{2} to the power of 2 and get \left(\frac{3a}{2}\right)^{2}.

$$\frac{145}{81}a^{2}-\frac{580}{81}a+\frac{580}{81}+9-3\times 3a+\left(\frac{3a}{2}\right)^{2}=34^{2}$$

Add \frac{580}{81} and 9 to get \frac{1309}{81}.

$$\frac{145}{81}a^{2}-\frac{580}{81}a+\frac{1309}{81}-3\times 3a+\left(\frac{3a}{2}\right)^{2}=34^{2}$$

Multiply -3 and 3 to get -9.

$$\frac{145}{81}a^{2}-\frac{580}{81}a+\frac{1309}{81}-9a+\left(\frac{3a}{2}\right)^{2}=34^{2}$$

Combine -\frac{580}{81}a and -9a to get -\frac{1309}{81}a.

$$\frac{145}{81}a^{2}-\frac{1309}{81}a+\frac{1309}{81}+\left(\frac{3a}{2}\right)^{2}=34^{2}$$

To raise \frac{3a}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{145}{81}a^{2}-\frac{1309}{81}a+\frac{1309}{81}+\frac{\left(3a\right)^{2}}{2^{2}}=34^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 81 and 2^{2} is 324. Multiply \frac{1309}{81} times \frac{4}{4}. Multiply \frac{\left(3a\right)^{2}}{2^{2}} times \frac{81}{81}.

$$\frac{145}{81}a^{2}-\frac{1309}{81}a+\frac{1309\times 4}{324}+\frac{81\times \left(3a\right)^{2}}{324}=34^{2}$$

Since \frac{1309\times 4}{324} and \frac{81\times \left(3a\right)^{2}}{324} have the same denominator, add them by adding their numerators.

$$\frac{145}{81}a^{2}-\frac{1309}{81}a+\frac{1309\times 4+81\times \left(3a\right)^{2}}{324}=34^{2}$$

Calculate 34 to the power of 2 and get 1156.

$$\frac{145}{81}a^{2}-\frac{1309}{81}a+\frac{1309\times 4+81\times \left(3a\right)^{2}}{324}=1156$$

Multiply 1309 and 4 to get 5236.

$$\frac{145}{81}a^{2}-\frac{1309}{81}a+\frac{5236+81\times \left(3a\right)^{2}}{324}=1156$$

Expand \left(3a\right)^{2}.

$$\frac{145}{81}a^{2}-\frac{1309}{81}a+\frac{5236+81\times 3^{2}a^{2}}{324}=1156$$

Calculate 3 to the power of 2 and get 9.

$$\frac{145}{81}a^{2}-\frac{1309}{81}a+\frac{5236+81\times 9a^{2}}{324}=1156$$

Multiply 81 and 9 to get 729.

$$\frac{145}{81}a^{2}-\frac{1309}{81}a+\frac{5236+729a^{2}}{324}=1156$$

Divide each term of 5236+729a^{2} by 324 to get \frac{1309}{81}+\frac{9}{4}a^{2}.

$$\frac{145}{81}a^{2}-\frac{1309}{81}a+\frac{1309}{81}+\frac{9}{4}a^{2}=1156$$

Combine \frac{145}{81}a^{2} and \frac{9}{4}a^{2} to get \frac{1309}{324}a^{2}.

$$\frac{1309}{324}a^{2}-\frac{1309}{81}a+\frac{1309}{81}=1156$$

Subtract 1156 from both sides.

$$\frac{1309}{324}a^{2}-\frac{1309}{81}a+\frac{1309}{81}-1156=0$$

Subtract 1156 from \frac{1309}{81} to get -\frac{92327}{81}.

$$\frac{1309}{324}a^{2}-\frac{1309}{81}a-\frac{92327}{81}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1309}{324} for a, -\frac{1309}{81} for b, and -\frac{92327}{81} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-\frac{1309}{81}\right)±\sqrt{\left(-\frac{1309}{81}\right)^{2}-4\times \frac{1309}{324}\left(-\frac{92327}{81}\right)}}{2\times \frac{1309}{324}}$$

Square -\frac{1309}{81} by squaring both the numerator and the denominator of the fraction.

$$a=\frac{-\left(-\frac{1309}{81}\right)±\sqrt{\frac{1713481}{6561}-4\times \frac{1309}{324}\left(-\frac{92327}{81}\right)}}{2\times \frac{1309}{324}}$$

Multiply -4 times \frac{1309}{324}.

$$a=\frac{-\left(-\frac{1309}{81}\right)±\sqrt{\frac{1713481}{6561}-\frac{1309}{81}\left(-\frac{92327}{81}\right)}}{2\times \frac{1309}{324}}$$

Multiply -\frac{1309}{81} times -\frac{92327}{81} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$a=\frac{-\left(-\frac{1309}{81}\right)±\sqrt{\frac{1713481+120856043}{6561}}}{2\times \frac{1309}{324}}$$

Add \frac{1713481}{6561} to \frac{120856043}{6561} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a=\frac{-\left(-\frac{1309}{81}\right)±\sqrt{\frac{1513204}{81}}}{2\times \frac{1309}{324}}$$

Take the square root of \frac{1513204}{81}.

$$a=\frac{-\left(-\frac{1309}{81}\right)±\frac{34\sqrt{1309}}{9}}{2\times \frac{1309}{324}}$$

The opposite of -\frac{1309}{81} is \frac{1309}{81}.

$$a=\frac{\frac{1309}{81}±\frac{34\sqrt{1309}}{9}}{2\times \frac{1309}{324}}$$

Multiply 2 times \frac{1309}{324}.

$$a=\frac{\frac{1309}{81}±\frac{34\sqrt{1309}}{9}}{\frac{1309}{162}}$$

Now solve the equation a=\frac{\frac{1309}{81}±\frac{34\sqrt{1309}}{9}}{\frac{1309}{162}} when ± is plus. Add \frac{1309}{81} to \frac{34\sqrt{1309}}{9}.

$$a=\frac{\frac{34\sqrt{1309}}{9}+\frac{1309}{81}}{\frac{1309}{162}}$$

Divide \frac{1309}{81}+\frac{34\sqrt{1309}}{9} by \frac{1309}{162} by multiplying \frac{1309}{81}+\frac{34\sqrt{1309}}{9} by the reciprocal of \frac{1309}{162}.

$$a=\frac{36\sqrt{1309}}{77}+2$$

Now solve the equation a=\frac{\frac{1309}{81}±\frac{34\sqrt{1309}}{9}}{\frac{1309}{162}} when ± is minus. Subtract \frac{34\sqrt{1309}}{9} from \frac{1309}{81}.

$$a=\frac{-\frac{34\sqrt{1309}}{9}+\frac{1309}{81}}{\frac{1309}{162}}$$

Divide \frac{1309}{81}-\frac{34\sqrt{1309}}{9} by \frac{1309}{162} by multiplying \frac{1309}{81}-\frac{34\sqrt{1309}}{9} by the reciprocal of \frac{1309}{162}.

$$a=-\frac{36\sqrt{1309}}{77}+2$$

The equation is now solved.

$$a=\frac{36\sqrt{1309}}{77}+2 a=-\frac{36\sqrt{1309}}{77}+2$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(a-2\right)^{2}.

$$a^{2}-4a+4+\left(\frac{8}{9}a-\frac{16}{9}\right)^{2}+\left(3-\frac{3a}{2}\right)^{2}=34^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{8}{9}a-\frac{16}{9}\right)^{2}.

$$a^{2}-4a+4+\frac{64}{81}a^{2}-\frac{256}{81}a+\frac{256}{81}+\left(3-\frac{3a}{2}\right)^{2}=34^{2}$$

Combine a^{2} and \frac{64}{81}a^{2} to get \frac{145}{81}a^{2}.

$$\frac{145}{81}a^{2}-4a+4-\frac{256}{81}a+\frac{256}{81}+\left(3-\frac{3a}{2}\right)^{2}=34^{2}$$

Combine -4a and -\frac{256}{81}a to get -\frac{580}{81}a.

$$\frac{145}{81}a^{2}-\frac{580}{81}a+4+\frac{256}{81}+\left(3-\frac{3a}{2}\right)^{2}=34^{2}$$

Add 4 and \frac{256}{81} to get \frac{580}{81}.

$$\frac{145}{81}a^{2}-\frac{580}{81}a+\frac{580}{81}+\left(3-\frac{3a}{2}\right)^{2}=34^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3-\frac{3a}{2}\right)^{2}.

$$\frac{145}{81}a^{2}-\frac{580}{81}a+\frac{580}{81}+9+6\left(-\frac{3a}{2}\right)+\left(-\frac{3a}{2}\right)^{2}=34^{2}$$

Cancel out 2, the greatest common factor in 6 and 2.

$$\frac{145}{81}a^{2}-\frac{580}{81}a+\frac{580}{81}+9-3\times 3a+\left(-\frac{3a}{2}\right)^{2}=34^{2}$$

Calculate -\frac{3a}{2} to the power of 2 and get \left(\frac{3a}{2}\right)^{2}.

$$\frac{145}{81}a^{2}-\frac{580}{81}a+\frac{580}{81}+9-3\times 3a+\left(\frac{3a}{2}\right)^{2}=34^{2}$$

Add \frac{580}{81} and 9 to get \frac{1309}{81}.

$$\frac{145}{81}a^{2}-\frac{580}{81}a+\frac{1309}{81}-3\times 3a+\left(\frac{3a}{2}\right)^{2}=34^{2}$$

Multiply -3 and 3 to get -9.

$$\frac{145}{81}a^{2}-\frac{580}{81}a+\frac{1309}{81}-9a+\left(\frac{3a}{2}\right)^{2}=34^{2}$$

Combine -\frac{580}{81}a and -9a to get -\frac{1309}{81}a.

$$\frac{145}{81}a^{2}-\frac{1309}{81}a+\frac{1309}{81}+\left(\frac{3a}{2}\right)^{2}=34^{2}$$

To raise \frac{3a}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{145}{81}a^{2}-\frac{1309}{81}a+\frac{1309}{81}+\frac{\left(3a\right)^{2}}{2^{2}}=34^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 81 and 2^{2} is 324. Multiply \frac{1309}{81} times \frac{4}{4}. Multiply \frac{\left(3a\right)^{2}}{2^{2}} times \frac{81}{81}.

$$\frac{145}{81}a^{2}-\frac{1309}{81}a+\frac{1309\times 4}{324}+\frac{81\times \left(3a\right)^{2}}{324}=34^{2}$$

Since \frac{1309\times 4}{324} and \frac{81\times \left(3a\right)^{2}}{324} have the same denominator, add them by adding their numerators.

$$\frac{145}{81}a^{2}-\frac{1309}{81}a+\frac{1309\times 4+81\times \left(3a\right)^{2}}{324}=34^{2}$$

Calculate 34 to the power of 2 and get 1156.

$$\frac{145}{81}a^{2}-\frac{1309}{81}a+\frac{1309\times 4+81\times \left(3a\right)^{2}}{324}=1156$$

Multiply 1309 and 4 to get 5236.

$$\frac{145}{81}a^{2}-\frac{1309}{81}a+\frac{5236+81\times \left(3a\right)^{2}}{324}=1156$$

Expand \left(3a\right)^{2}.

$$\frac{145}{81}a^{2}-\frac{1309}{81}a+\frac{5236+81\times 3^{2}a^{2}}{324}=1156$$

Calculate 3 to the power of 2 and get 9.

$$\frac{145}{81}a^{2}-\frac{1309}{81}a+\frac{5236+81\times 9a^{2}}{324}=1156$$

Multiply 81 and 9 to get 729.

$$\frac{145}{81}a^{2}-\frac{1309}{81}a+\frac{5236+729a^{2}}{324}=1156$$

Divide each term of 5236+729a^{2} by 324 to get \frac{1309}{81}+\frac{9}{4}a^{2}.

$$\frac{145}{81}a^{2}-\frac{1309}{81}a+\frac{1309}{81}+\frac{9}{4}a^{2}=1156$$

Combine \frac{145}{81}a^{2} and \frac{9}{4}a^{2} to get \frac{1309}{324}a^{2}.

$$\frac{1309}{324}a^{2}-\frac{1309}{81}a+\frac{1309}{81}=1156$$

Subtract \frac{1309}{81} from both sides.

$$\frac{1309}{324}a^{2}-\frac{1309}{81}a=1156-\frac{1309}{81}$$

Subtract \frac{1309}{81} from 1156 to get \frac{92327}{81}.

$$\frac{1309}{324}a^{2}-\frac{1309}{81}a=\frac{92327}{81}$$

Divide both sides of the equation by \frac{1309}{324}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{1309}{324}a^{2}-\frac{1309}{81}a}{\frac{1309}{324}}=\frac{\frac{92327}{81}}{\frac{1309}{324}}$$

Dividing by \frac{1309}{324} undoes the multiplication by \frac{1309}{324}.

$$a^{2}+\left(-\frac{\frac{1309}{81}}{\frac{1309}{324}}\right)a=\frac{\frac{92327}{81}}{\frac{1309}{324}}$$

Divide -\frac{1309}{81} by \frac{1309}{324} by multiplying -\frac{1309}{81} by the reciprocal of \frac{1309}{324}.

$$a^{2}-4a=\frac{\frac{92327}{81}}{\frac{1309}{324}}$$

Divide \frac{92327}{81} by \frac{1309}{324} by multiplying \frac{92327}{81} by the reciprocal of \frac{1309}{324}.

$$a^{2}-4a=\frac{21724}{77}$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-4a+\left(-2\right)^{2}=\frac{21724}{77}+\left(-2\right)^{2}$$

Square -2.

$$a^{2}-4a+4=\frac{21724}{77}+4$$

Add \frac{21724}{77} to 4.

$$a^{2}-4a+4=\frac{22032}{77}$$

Factor a^{2}-4a+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-2\right)^{2}=\frac{22032}{77}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-2\right)^{2}}=\sqrt{\frac{22032}{77}}$$

Simplify.

$$a-2=\frac{36\sqrt{1309}}{77} a-2=-\frac{36\sqrt{1309}}{77}$$

Add 2 to both sides of the equation.

$$a=\frac{36\sqrt{1309}}{77}+2 a=-\frac{36\sqrt{1309}}{77}+2$$