Solve (3a-6)^2/49=(9a+73)^2/5329

$$\frac { ( 3 a - 6 ) ^ { 2 } } { 49 } = \frac { ( 9 a + 73 ) ^ { 2 } } { 5329 }$$

Solve for a

$$a=-\frac{73}{282}\approx -0.258865248$$

$$a = \frac{73}{12} = 6\frac{1}{12} \approx 6.083333333$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 261121, the least common multiple of 49,5329.

$$5329\left(3a-6\right)^{2}=49\left(9a+73\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3a-6\right)^{2}.

$$5329\left(9a^{2}-36a+36\right)=49\left(9a+73\right)^{2}$$

Use the distributive property to multiply 5329 by 9a^{2}-36a+36.

$$47961a^{2}-191844a+191844=49\left(9a+73\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(9a+73\right)^{2}.

$$47961a^{2}-191844a+191844=49\left(81a^{2}+1314a+5329\right)$$

Use the distributive property to multiply 49 by 81a^{2}+1314a+5329.

$$47961a^{2}-191844a+191844=3969a^{2}+64386a+261121$$

Subtract 3969a^{2} from both sides.

$$47961a^{2}-191844a+191844-3969a^{2}=64386a+261121$$

Combine 47961a^{2} and -3969a^{2} to get 43992a^{2}.

$$43992a^{2}-191844a+191844=64386a+261121$$

Subtract 64386a from both sides.

$$43992a^{2}-191844a+191844-64386a=261121$$

Combine -191844a and -64386a to get -256230a.

$$43992a^{2}-256230a+191844=261121$$

Subtract 261121 from both sides.

$$43992a^{2}-256230a+191844-261121=0$$

Subtract 261121 from 191844 to get -69277.

$$43992a^{2}-256230a-69277=0$$

Divide both sides by 13.

$$3384a^{2}-19710a-5329=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 3384a^{2}+aa+ba-5329. To find a and b, set up a system to be solved.

$$a+b=-19710 ab=3384\left(-5329\right)=-18033336$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -18033336.

$$1,-18033336 2,-9016668 3,-6011112 4,-4508334 6,-3005556 8,-2254167 9,-2003704 12,-1502778 18,-1001852 24,-751389 36,-500926 47,-383688 72,-250463 73,-247032 94,-191844 141,-127896 146,-123516 188,-95922 219,-82344 282,-63948 292,-61758 376,-47961 423,-42632 438,-41172 564,-31974 584,-30879 657,-27448 846,-21316 876,-20586 1128,-15987 1314,-13724 1692,-10658 1752,-10293 2628,-6862 3384,-5329 3431,-5256$$

Calculate the sum for each pair.

$$1-18033336=-18033335 2-9016668=-9016666 3-6011112=-6011109 4-4508334=-4508330 6-3005556=-3005550 8-2254167=-2254159 9-2003704=-2003695 12-1502778=-1502766 18-1001852=-1001834 24-751389=-751365 36-500926=-500890 47-383688=-383641 72-250463=-250391 73-247032=-246959 94-191844=-191750 141-127896=-127755 146-123516=-123370 188-95922=-95734 219-82344=-82125 282-63948=-63666 292-61758=-61466 376-47961=-47585 423-42632=-42209 438-41172=-40734 564-31974=-31410 584-30879=-30295 657-27448=-26791 846-21316=-20470 876-20586=-19710 1128-15987=-14859 1314-13724=-12410 1692-10658=-8966 1752-10293=-8541 2628-6862=-4234 3384-5329=-1945 3431-5256=-1825$$

The solution is the pair that gives sum -19710.

$$a=-20586 b=876$$

Rewrite 3384a^{2}-19710a-5329 as \left(3384a^{2}-20586a\right)+\left(876a-5329\right).

$$\left(3384a^{2}-20586a\right)+\left(876a-5329\right)$$

Factor out 282a in the first and 73 in the second group.

$$282a\left(12a-73\right)+73\left(12a-73\right)$$

Factor out common term 12a-73 by using distributive property.

$$\left(12a-73\right)\left(282a+73\right)$$

To find equation solutions, solve 12a-73=0 and 282a+73=0.

$$a=\frac{73}{12} a=-\frac{73}{282}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 261121, the least common multiple of 49,5329.

$$5329\left(3a-6\right)^{2}=49\left(9a+73\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3a-6\right)^{2}.

$$5329\left(9a^{2}-36a+36\right)=49\left(9a+73\right)^{2}$$

Use the distributive property to multiply 5329 by 9a^{2}-36a+36.

$$47961a^{2}-191844a+191844=49\left(9a+73\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(9a+73\right)^{2}.

$$47961a^{2}-191844a+191844=49\left(81a^{2}+1314a+5329\right)$$

Use the distributive property to multiply 49 by 81a^{2}+1314a+5329.

$$47961a^{2}-191844a+191844=3969a^{2}+64386a+261121$$

Subtract 3969a^{2} from both sides.

$$47961a^{2}-191844a+191844-3969a^{2}=64386a+261121$$

Combine 47961a^{2} and -3969a^{2} to get 43992a^{2}.

$$43992a^{2}-191844a+191844=64386a+261121$$

Subtract 64386a from both sides.

$$43992a^{2}-191844a+191844-64386a=261121$$

Combine -191844a and -64386a to get -256230a.

$$43992a^{2}-256230a+191844=261121$$

Subtract 261121 from both sides.

$$43992a^{2}-256230a+191844-261121=0$$

Subtract 261121 from 191844 to get -69277.

$$43992a^{2}-256230a-69277=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 43992 for a, -256230 for b, and -69277 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-256230\right)±\sqrt{\left(-256230\right)^{2}-4\times 43992\left(-69277\right)}}{2\times 43992}$$

Square -256230.

$$a=\frac{-\left(-256230\right)±\sqrt{65653812900-4\times 43992\left(-69277\right)}}{2\times 43992}$$

Multiply -4 times 43992.

$$a=\frac{-\left(-256230\right)±\sqrt{65653812900-175968\left(-69277\right)}}{2\times 43992}$$

Multiply -175968 times -69277.

$$a=\frac{-\left(-256230\right)±\sqrt{65653812900+12190535136}}{2\times 43992}$$

Add 65653812900 to 12190535136.

$$a=\frac{-\left(-256230\right)±\sqrt{77844348036}}{2\times 43992}$$

Take the square root of 77844348036.

$$a=\frac{-\left(-256230\right)±279006}{2\times 43992}$$

The opposite of -256230 is 256230.

$$a=\frac{256230±279006}{2\times 43992}$$

Multiply 2 times 43992.

$$a=\frac{256230±279006}{87984}$$

Now solve the equation a=\frac{256230±279006}{87984} when ± is plus. Add 256230 to 279006.

$$a=\frac{535236}{87984}$$

Reduce the fraction \frac{535236}{87984} to lowest terms by extracting and canceling out 7332.

$$a=\frac{73}{12}$$

Now solve the equation a=\frac{256230±279006}{87984} when ± is minus. Subtract 279006 from 256230.

$$a=-\frac{22776}{87984}$$

Reduce the fraction \frac{-22776}{87984} to lowest terms by extracting and canceling out 312.

$$a=-\frac{73}{282}$$

The equation is now solved.

$$a=\frac{73}{12} a=-\frac{73}{282}$$

Steps for Completing the Square

Multiply both sides of the equation by 261121, the least common multiple of 49,5329.

$$5329\left(3a-6\right)^{2}=49\left(9a+73\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3a-6\right)^{2}.

$$5329\left(9a^{2}-36a+36\right)=49\left(9a+73\right)^{2}$$

Use the distributive property to multiply 5329 by 9a^{2}-36a+36.

$$47961a^{2}-191844a+191844=49\left(9a+73\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(9a+73\right)^{2}.

$$47961a^{2}-191844a+191844=49\left(81a^{2}+1314a+5329\right)$$

Use the distributive property to multiply 49 by 81a^{2}+1314a+5329.

$$47961a^{2}-191844a+191844=3969a^{2}+64386a+261121$$

Subtract 3969a^{2} from both sides.

$$47961a^{2}-191844a+191844-3969a^{2}=64386a+261121$$

Combine 47961a^{2} and -3969a^{2} to get 43992a^{2}.

$$43992a^{2}-191844a+191844=64386a+261121$$

Subtract 64386a from both sides.

$$43992a^{2}-191844a+191844-64386a=261121$$

Combine -191844a and -64386a to get -256230a.

$$43992a^{2}-256230a+191844=261121$$

Subtract 191844 from both sides.

$$43992a^{2}-256230a=261121-191844$$

Subtract 191844 from 261121 to get 69277.

$$43992a^{2}-256230a=69277$$

Divide both sides by 43992.

$$\frac{43992a^{2}-256230a}{43992}=\frac{69277}{43992}$$

Dividing by 43992 undoes the multiplication by 43992.

$$a^{2}+\left(-\frac{256230}{43992}\right)a=\frac{69277}{43992}$$

Reduce the fraction \frac{-256230}{43992} to lowest terms by extracting and canceling out 234.

$$a^{2}-\frac{1095}{188}a=\frac{69277}{43992}$$

Reduce the fraction \frac{69277}{43992} to lowest terms by extracting and canceling out 13.

$$a^{2}-\frac{1095}{188}a=\frac{5329}{3384}$$

Divide -\frac{1095}{188}, the coefficient of the x term, by 2 to get -\frac{1095}{376}. Then add the square of -\frac{1095}{376} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{1095}{188}a+\left(-\frac{1095}{376}\right)^{2}=\frac{5329}{3384}+\left(-\frac{1095}{376}\right)^{2}$$

Square -\frac{1095}{376} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{1095}{188}a+\frac{1199025}{141376}=\frac{5329}{3384}+\frac{1199025}{141376}$$

Add \frac{5329}{3384} to \frac{1199025}{141376} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}-\frac{1095}{188}a+\frac{1199025}{141376}=\frac{12794929}{1272384}$$

Factor a^{2}-\frac{1095}{188}a+\frac{1199025}{141376}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{1095}{376}\right)^{2}=\frac{12794929}{1272384}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{1095}{376}\right)^{2}}=\sqrt{\frac{12794929}{1272384}}$$

Simplify.

$$a-\frac{1095}{376}=\frac{3577}{1128} a-\frac{1095}{376}=-\frac{3577}{1128}$$

Add \frac{1095}{376} to both sides of the equation.

$$a=\frac{73}{12} a=-\frac{73}{282}$$