Solve 4-3x/x+1+x+1/4-3x=50/7

$$\frac { 4 - 3 x } { x + 1 } + \frac { x + 1 } { 4 - 3 x } = \frac { 50 } { 7 }$$

Solve for x

$$x=-\frac{3}{10}=-0.3$$

$$x = \frac{27}{22} = 1\frac{5}{22} \approx 1.227272727$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,\frac{4}{3} since division by zero is not defined. Multiply both sides of the equation by 7\left(3x-4\right)\left(x+1\right), the least common multiple of x+1,4-3x,7.

$$\left(21x-28\right)\left(4-3x\right)+\left(-7-7x\right)\left(x+1\right)=50\left(3x-4\right)\left(x+1\right)$$

Use the distributive property to multiply 21x-28 by 4-3x and combine like terms.

$$168x-63x^{2}-112+\left(-7-7x\right)\left(x+1\right)=50\left(3x-4\right)\left(x+1\right)$$

Use the distributive property to multiply -7-7x by x+1 and combine like terms.

$$168x-63x^{2}-112-14x-7-7x^{2}=50\left(3x-4\right)\left(x+1\right)$$

Combine 168x and -14x to get 154x.

$$154x-63x^{2}-112-7-7x^{2}=50\left(3x-4\right)\left(x+1\right)$$

Subtract 7 from -112 to get -119.

$$154x-63x^{2}-119-7x^{2}=50\left(3x-4\right)\left(x+1\right)$$

Combine -63x^{2} and -7x^{2} to get -70x^{2}.

$$154x-70x^{2}-119=50\left(3x-4\right)\left(x+1\right)$$

Use the distributive property to multiply 50 by 3x-4.

$$154x-70x^{2}-119=\left(150x-200\right)\left(x+1\right)$$

Use the distributive property to multiply 150x-200 by x+1 and combine like terms.

$$154x-70x^{2}-119=150x^{2}-50x-200$$

Subtract 150x^{2} from both sides.

$$154x-70x^{2}-119-150x^{2}=-50x-200$$

Combine -70x^{2} and -150x^{2} to get -220x^{2}.

$$154x-220x^{2}-119=-50x-200$$

Add 50x to both sides.

$$154x-220x^{2}-119+50x=-200$$

Combine 154x and 50x to get 204x.

$$204x-220x^{2}-119=-200$$

Add 200 to both sides.

$$204x-220x^{2}-119+200=0$$

Add -119 and 200 to get 81.

$$204x-220x^{2}+81=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-220x^{2}+204x+81=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -220 for a, 204 for b, and 81 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-204±\sqrt{204^{2}-4\left(-220\right)\times 81}}{2\left(-220\right)}$$

Square 204.

$$x=\frac{-204±\sqrt{41616-4\left(-220\right)\times 81}}{2\left(-220\right)}$$

Multiply -4 times -220.

$$x=\frac{-204±\sqrt{41616+880\times 81}}{2\left(-220\right)}$$

Multiply 880 times 81.

$$x=\frac{-204±\sqrt{41616+71280}}{2\left(-220\right)}$$

Add 41616 to 71280.

$$x=\frac{-204±\sqrt{112896}}{2\left(-220\right)}$$

Take the square root of 112896.

$$x=\frac{-204±336}{2\left(-220\right)}$$

Multiply 2 times -220.

$$x=\frac{-204±336}{-440}$$

Now solve the equation x=\frac{-204±336}{-440} when ± is plus. Add -204 to 336.

$$x=\frac{132}{-440}$$

Reduce the fraction \frac{132}{-440} to lowest terms by extracting and canceling out 44.

$$x=-\frac{3}{10}$$

Now solve the equation x=\frac{-204±336}{-440} when ± is minus. Subtract 336 from -204.

$$x=-\frac{540}{-440}$$

Reduce the fraction \frac{-540}{-440} to lowest terms by extracting and canceling out 20.

$$x=\frac{27}{22}$$

The equation is now solved.

$$x=-\frac{3}{10} x=\frac{27}{22}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,\frac{4}{3} since division by zero is not defined. Multiply both sides of the equation by 7\left(3x-4\right)\left(x+1\right), the least common multiple of x+1,4-3x,7.

$$\left(21x-28\right)\left(4-3x\right)+\left(-7-7x\right)\left(x+1\right)=50\left(3x-4\right)\left(x+1\right)$$

Use the distributive property to multiply 21x-28 by 4-3x and combine like terms.

$$168x-63x^{2}-112+\left(-7-7x\right)\left(x+1\right)=50\left(3x-4\right)\left(x+1\right)$$

Use the distributive property to multiply -7-7x by x+1 and combine like terms.

$$168x-63x^{2}-112-14x-7-7x^{2}=50\left(3x-4\right)\left(x+1\right)$$

Combine 168x and -14x to get 154x.

$$154x-63x^{2}-112-7-7x^{2}=50\left(3x-4\right)\left(x+1\right)$$

Subtract 7 from -112 to get -119.

$$154x-63x^{2}-119-7x^{2}=50\left(3x-4\right)\left(x+1\right)$$

Combine -63x^{2} and -7x^{2} to get -70x^{2}.

$$154x-70x^{2}-119=50\left(3x-4\right)\left(x+1\right)$$

Use the distributive property to multiply 50 by 3x-4.

$$154x-70x^{2}-119=\left(150x-200\right)\left(x+1\right)$$

Use the distributive property to multiply 150x-200 by x+1 and combine like terms.

$$154x-70x^{2}-119=150x^{2}-50x-200$$

Subtract 150x^{2} from both sides.

$$154x-70x^{2}-119-150x^{2}=-50x-200$$

Combine -70x^{2} and -150x^{2} to get -220x^{2}.

$$154x-220x^{2}-119=-50x-200$$

Add 50x to both sides.

$$154x-220x^{2}-119+50x=-200$$

Combine 154x and 50x to get 204x.

$$204x-220x^{2}-119=-200$$

Add 119 to both sides.

$$204x-220x^{2}=-200+119$$

Add -200 and 119 to get -81.

$$204x-220x^{2}=-81$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-220x^{2}+204x=-81$$

Divide both sides by -220.

$$\frac{-220x^{2}+204x}{-220}=-\frac{81}{-220}$$

Dividing by -220 undoes the multiplication by -220.

$$x^{2}+\frac{204}{-220}x=-\frac{81}{-220}$$

Reduce the fraction \frac{204}{-220} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{51}{55}x=-\frac{81}{-220}$$

Divide -81 by -220.

$$x^{2}-\frac{51}{55}x=\frac{81}{220}$$

Divide -\frac{51}{55}, the coefficient of the x term, by 2 to get -\frac{51}{110}. Then add the square of -\frac{51}{110} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{51}{55}x+\left(-\frac{51}{110}\right)^{2}=\frac{81}{220}+\left(-\frac{51}{110}\right)^{2}$$

Square -\frac{51}{110} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{51}{55}x+\frac{2601}{12100}=\frac{81}{220}+\frac{2601}{12100}$$

Add \frac{81}{220} to \frac{2601}{12100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{51}{55}x+\frac{2601}{12100}=\frac{1764}{3025}$$

Factor x^{2}-\frac{51}{55}x+\frac{2601}{12100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{51}{110}\right)^{2}=\frac{1764}{3025}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{51}{110}\right)^{2}}=\sqrt{\frac{1764}{3025}}$$

Simplify.

$$x-\frac{51}{110}=\frac{42}{55} x-\frac{51}{110}=-\frac{42}{55}$$

Add \frac{51}{110} to both sides of the equation.

$$x=\frac{27}{22} x=-\frac{3}{10}$$