Solve x-43/57*x+46/54=x+49/51+x+52/48

$$\frac { x - 43 } { 57 } \times \frac { x + 46 } { 54 } = \frac { x + 49 } { 51 } + \frac { x + 52 } { 48 }$$

Solve for x

$$x = \frac{\sqrt{884775889} + 16521}{272} \approx 170.096259626$$

$$x=\frac{16521-\sqrt{884775889}}{272}\approx -48.61831845$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 139536, the least common multiple of 57,54,51,48.

$$2448\left(x-43\right)\times \frac{x+46}{54}=2736\left(x+49\right)+2907\left(x+52\right)$$

Express 2448\times \frac{x+46}{54} as a single fraction.

$$\frac{2448\left(x+46\right)}{54}\left(x-43\right)=2736\left(x+49\right)+2907\left(x+52\right)$$

Use the distributive property to multiply \frac{2448\left(x+46\right)}{54} by x-43.

$$\frac{2448\left(x+46\right)}{54}x-43\times \frac{2448\left(x+46\right)}{54}=2736\left(x+49\right)+2907\left(x+52\right)$$

Use the distributive property to multiply 2448 by x+46.

$$\frac{2448x+112608}{54}x-43\times \frac{2448\left(x+46\right)}{54}=2736\left(x+49\right)+2907\left(x+52\right)$$

Express \frac{2448x+112608}{54}x as a single fraction.

$$\frac{\left(2448x+112608\right)x}{54}-43\times \frac{2448\left(x+46\right)}{54}=2736\left(x+49\right)+2907\left(x+52\right)$$

Use the distributive property to multiply 2448 by x+46.

$$\frac{\left(2448x+112608\right)x}{54}-43\times \frac{2448x+112608}{54}=2736\left(x+49\right)+2907\left(x+52\right)$$

Express -43\times \frac{2448x+112608}{54} as a single fraction.

$$\frac{\left(2448x+112608\right)x}{54}+\frac{-43\left(2448x+112608\right)}{54}=2736\left(x+49\right)+2907\left(x+52\right)$$

Since \frac{\left(2448x+112608\right)x}{54} and \frac{-43\left(2448x+112608\right)}{54} have the same denominator, add them by adding their numerators.

$$\frac{\left(2448x+112608\right)x-43\left(2448x+112608\right)}{54}=2736\left(x+49\right)+2907\left(x+52\right)$$

Do the multiplications in \left(2448x+112608\right)x-43\left(2448x+112608\right).

$$\frac{2448x^{2}+112608x-105264x-4842144}{54}=2736\left(x+49\right)+2907\left(x+52\right)$$

Combine like terms in 2448x^{2}+112608x-105264x-4842144.

$$\frac{2448x^{2}+7344x-4842144}{54}=2736\left(x+49\right)+2907\left(x+52\right)$$

Use the distributive property to multiply 2736 by x+49.

$$\frac{2448x^{2}+7344x-4842144}{54}=2736x+134064+2907\left(x+52\right)$$

Use the distributive property to multiply 2907 by x+52.

$$\frac{2448x^{2}+7344x-4842144}{54}=2736x+134064+2907x+151164$$

Combine 2736x and 2907x to get 5643x.

$$\frac{2448x^{2}+7344x-4842144}{54}=5643x+134064+151164$$

Add 134064 and 151164 to get 285228.

$$\frac{2448x^{2}+7344x-4842144}{54}=5643x+285228$$

Divide each term of 2448x^{2}+7344x-4842144 by 54 to get \frac{136}{3}x^{2}+136x-\frac{269008}{3}.

$$\frac{136}{3}x^{2}+136x-\frac{269008}{3}=5643x+285228$$

Subtract 5643x from both sides.

$$\frac{136}{3}x^{2}+136x-\frac{269008}{3}-5643x=285228$$

Combine 136x and -5643x to get -5507x.

$$\frac{136}{3}x^{2}-5507x-\frac{269008}{3}=285228$$

Subtract 285228 from both sides.

$$\frac{136}{3}x^{2}-5507x-\frac{269008}{3}-285228=0$$

Convert 285228 to fraction \frac{855684}{3}.

$$\frac{136}{3}x^{2}-5507x-\frac{269008}{3}-\frac{855684}{3}=0$$

Since -\frac{269008}{3} and \frac{855684}{3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{136}{3}x^{2}-5507x+\frac{-269008-855684}{3}=0$$

Subtract 855684 from -269008 to get -1124692.

$$\frac{136}{3}x^{2}-5507x-\frac{1124692}{3}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{136}{3} for a, -5507 for b, and -\frac{1124692}{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5507\right)±\sqrt{\left(-5507\right)^{2}-4\times \frac{136}{3}\left(-\frac{1124692}{3}\right)}}{2\times \frac{136}{3}}$$

Square -5507.

$$x=\frac{-\left(-5507\right)±\sqrt{30327049-4\times \frac{136}{3}\left(-\frac{1124692}{3}\right)}}{2\times \frac{136}{3}}$$

Multiply -4 times \frac{136}{3}.

$$x=\frac{-\left(-5507\right)±\sqrt{30327049-\frac{544}{3}\left(-\frac{1124692}{3}\right)}}{2\times \frac{136}{3}}$$

Multiply -\frac{544}{3} times -\frac{1124692}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-5507\right)±\sqrt{30327049+\frac{611832448}{9}}}{2\times \frac{136}{3}}$$

Add 30327049 to \frac{611832448}{9}.

$$x=\frac{-\left(-5507\right)±\sqrt{\frac{884775889}{9}}}{2\times \frac{136}{3}}$$

Take the square root of \frac{884775889}{9}.

$$x=\frac{-\left(-5507\right)±\frac{\sqrt{884775889}}{3}}{2\times \frac{136}{3}}$$

The opposite of -5507 is 5507.

$$x=\frac{5507±\frac{\sqrt{884775889}}{3}}{2\times \frac{136}{3}}$$

Multiply 2 times \frac{136}{3}.

$$x=\frac{5507±\frac{\sqrt{884775889}}{3}}{\frac{272}{3}}$$

Now solve the equation x=\frac{5507±\frac{\sqrt{884775889}}{3}}{\frac{272}{3}} when ± is plus. Add 5507 to \frac{\sqrt{884775889}}{3}.

$$x=\frac{\frac{\sqrt{884775889}}{3}+5507}{\frac{272}{3}}$$

Divide 5507+\frac{\sqrt{884775889}}{3} by \frac{272}{3} by multiplying 5507+\frac{\sqrt{884775889}}{3} by the reciprocal of \frac{272}{3}.

$$x=\frac{\sqrt{884775889}+16521}{272}$$

Now solve the equation x=\frac{5507±\frac{\sqrt{884775889}}{3}}{\frac{272}{3}} when ± is minus. Subtract \frac{\sqrt{884775889}}{3} from 5507.

$$x=\frac{-\frac{\sqrt{884775889}}{3}+5507}{\frac{272}{3}}$$

Divide 5507-\frac{\sqrt{884775889}}{3} by \frac{272}{3} by multiplying 5507-\frac{\sqrt{884775889}}{3} by the reciprocal of \frac{272}{3}.

$$x=\frac{16521-\sqrt{884775889}}{272}$$

The equation is now solved.

$$x=\frac{\sqrt{884775889}+16521}{272} x=\frac{16521-\sqrt{884775889}}{272}$$

Steps for Completing the Square

Multiply both sides of the equation by 139536, the least common multiple of 57,54,51,48.

$$2448\left(x-43\right)\times \frac{x+46}{54}=2736\left(x+49\right)+2907\left(x+52\right)$$

Express 2448\times \frac{x+46}{54} as a single fraction.

$$\frac{2448\left(x+46\right)}{54}\left(x-43\right)=2736\left(x+49\right)+2907\left(x+52\right)$$

Use the distributive property to multiply \frac{2448\left(x+46\right)}{54} by x-43.

$$\frac{2448\left(x+46\right)}{54}x-43\times \frac{2448\left(x+46\right)}{54}=2736\left(x+49\right)+2907\left(x+52\right)$$

Use the distributive property to multiply 2448 by x+46.

$$\frac{2448x+112608}{54}x-43\times \frac{2448\left(x+46\right)}{54}=2736\left(x+49\right)+2907\left(x+52\right)$$

Express \frac{2448x+112608}{54}x as a single fraction.

$$\frac{\left(2448x+112608\right)x}{54}-43\times \frac{2448\left(x+46\right)}{54}=2736\left(x+49\right)+2907\left(x+52\right)$$

Use the distributive property to multiply 2448 by x+46.

$$\frac{\left(2448x+112608\right)x}{54}-43\times \frac{2448x+112608}{54}=2736\left(x+49\right)+2907\left(x+52\right)$$

Express -43\times \frac{2448x+112608}{54} as a single fraction.

$$\frac{\left(2448x+112608\right)x}{54}+\frac{-43\left(2448x+112608\right)}{54}=2736\left(x+49\right)+2907\left(x+52\right)$$

Since \frac{\left(2448x+112608\right)x}{54} and \frac{-43\left(2448x+112608\right)}{54} have the same denominator, add them by adding their numerators.

$$\frac{\left(2448x+112608\right)x-43\left(2448x+112608\right)}{54}=2736\left(x+49\right)+2907\left(x+52\right)$$

Do the multiplications in \left(2448x+112608\right)x-43\left(2448x+112608\right).

$$\frac{2448x^{2}+112608x-105264x-4842144}{54}=2736\left(x+49\right)+2907\left(x+52\right)$$

Combine like terms in 2448x^{2}+112608x-105264x-4842144.

$$\frac{2448x^{2}+7344x-4842144}{54}=2736\left(x+49\right)+2907\left(x+52\right)$$

Use the distributive property to multiply 2736 by x+49.

$$\frac{2448x^{2}+7344x-4842144}{54}=2736x+134064+2907\left(x+52\right)$$

Use the distributive property to multiply 2907 by x+52.

$$\frac{2448x^{2}+7344x-4842144}{54}=2736x+134064+2907x+151164$$

Combine 2736x and 2907x to get 5643x.

$$\frac{2448x^{2}+7344x-4842144}{54}=5643x+134064+151164$$

Add 134064 and 151164 to get 285228.

$$\frac{2448x^{2}+7344x-4842144}{54}=5643x+285228$$

Divide each term of 2448x^{2}+7344x-4842144 by 54 to get \frac{136}{3}x^{2}+136x-\frac{269008}{3}.

$$\frac{136}{3}x^{2}+136x-\frac{269008}{3}=5643x+285228$$

Subtract 5643x from both sides.

$$\frac{136}{3}x^{2}+136x-\frac{269008}{3}-5643x=285228$$

Combine 136x and -5643x to get -5507x.

$$\frac{136}{3}x^{2}-5507x-\frac{269008}{3}=285228$$

Add \frac{269008}{3} to both sides.

$$\frac{136}{3}x^{2}-5507x=285228+\frac{269008}{3}$$

Convert 285228 to fraction \frac{855684}{3}.

$$\frac{136}{3}x^{2}-5507x=\frac{855684}{3}+\frac{269008}{3}$$

Since \frac{855684}{3} and \frac{269008}{3} have the same denominator, add them by adding their numerators.

$$\frac{136}{3}x^{2}-5507x=\frac{855684+269008}{3}$$

Add 855684 and 269008 to get 1124692.

$$\frac{136}{3}x^{2}-5507x=\frac{1124692}{3}$$

Divide both sides of the equation by \frac{136}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{136}{3}x^{2}-5507x}{\frac{136}{3}}=\frac{\frac{1124692}{3}}{\frac{136}{3}}$$

Dividing by \frac{136}{3} undoes the multiplication by \frac{136}{3}.

$$x^{2}+\left(-\frac{5507}{\frac{136}{3}}\right)x=\frac{\frac{1124692}{3}}{\frac{136}{3}}$$

Divide -5507 by \frac{136}{3} by multiplying -5507 by the reciprocal of \frac{136}{3}.

$$x^{2}-\frac{16521}{136}x=\frac{\frac{1124692}{3}}{\frac{136}{3}}$$

Divide \frac{1124692}{3} by \frac{136}{3} by multiplying \frac{1124692}{3} by the reciprocal of \frac{136}{3}.

$$x^{2}-\frac{16521}{136}x=\frac{281173}{34}$$

Divide -\frac{16521}{136}, the coefficient of the x term, by 2 to get -\frac{16521}{272}. Then add the square of -\frac{16521}{272} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{16521}{136}x+\left(-\frac{16521}{272}\right)^{2}=\frac{281173}{34}+\left(-\frac{16521}{272}\right)^{2}$$

Square -\frac{16521}{272} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{16521}{136}x+\frac{272943441}{73984}=\frac{281173}{34}+\frac{272943441}{73984}$$

Add \frac{281173}{34} to \frac{272943441}{73984} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{16521}{136}x+\frac{272943441}{73984}=\frac{884775889}{73984}$$

Factor x^{2}-\frac{16521}{136}x+\frac{272943441}{73984}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{16521}{272}\right)^{2}=\frac{884775889}{73984}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{16521}{272}\right)^{2}}=\sqrt{\frac{884775889}{73984}}$$

Simplify.

$$x-\frac{16521}{272}=\frac{\sqrt{884775889}}{272} x-\frac{16521}{272}=-\frac{\sqrt{884775889}}{272}$$

Add \frac{16521}{272} to both sides of the equation.

$$x=\frac{\sqrt{884775889}+16521}{272} x=\frac{16521-\sqrt{884775889}}{272}$$