Solve x+7/x-1+x-3/3x+2=2-9x^2+31/3x^2-x-2

$$\frac { x + 7 } { x - 1 } + \frac { x - 3 } { 3 x + 2 } = 2 - \frac { 9 x ^ { 2 } + 31 } { 3 x ^ { 2 } - x - 2 }$$

Solve for x (complex solution)

$$x=\frac{\sqrt{1015}i}{14}-\frac{3}{2}\approx -1.5+2.275647475i$$

$$x=-\frac{\sqrt{1015}i}{14}-\frac{3}{2}\approx -1.5-2.275647475i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{2}{3},1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(3x+2\right), the least common multiple of x-1,3x+2,3x^{2}-x-2.

$$\left(3x+2\right)\left(x+7\right)+\left(x-1\right)\left(x-3\right)=\left(x-1\right)\left(3x+2\right)\times 2-\left(9x^{2}+31\right)$$

Use the distributive property to multiply 3x+2 by x+7 and combine like terms.

$$3x^{2}+23x+14+\left(x-1\right)\left(x-3\right)=\left(x-1\right)\left(3x+2\right)\times 2-\left(9x^{2}+31\right)$$

Use the distributive property to multiply x-1 by x-3 and combine like terms.

$$3x^{2}+23x+14+x^{2}-4x+3=\left(x-1\right)\left(3x+2\right)\times 2-\left(9x^{2}+31\right)$$

Combine 3x^{2} and x^{2} to get 4x^{2}.

$$4x^{2}+23x+14-4x+3=\left(x-1\right)\left(3x+2\right)\times 2-\left(9x^{2}+31\right)$$

Combine 23x and -4x to get 19x.

$$4x^{2}+19x+14+3=\left(x-1\right)\left(3x+2\right)\times 2-\left(9x^{2}+31\right)$$

Add 14 and 3 to get 17.

$$4x^{2}+19x+17=\left(x-1\right)\left(3x+2\right)\times 2-\left(9x^{2}+31\right)$$

Use the distributive property to multiply x-1 by 3x+2 and combine like terms.

$$4x^{2}+19x+17=\left(3x^{2}-x-2\right)\times 2-\left(9x^{2}+31\right)$$

Use the distributive property to multiply 3x^{2}-x-2 by 2.

$$4x^{2}+19x+17=6x^{2}-2x-4-\left(9x^{2}+31\right)$$

To find the opposite of 9x^{2}+31, find the opposite of each term.

$$4x^{2}+19x+17=6x^{2}-2x-4-9x^{2}-31$$

Combine 6x^{2} and -9x^{2} to get -3x^{2}.

$$4x^{2}+19x+17=-3x^{2}-2x-4-31$$

Subtract 31 from -4 to get -35.

$$4x^{2}+19x+17=-3x^{2}-2x-35$$

Add 3x^{2} to both sides.

$$4x^{2}+19x+17+3x^{2}=-2x-35$$

Combine 4x^{2} and 3x^{2} to get 7x^{2}.

$$7x^{2}+19x+17=-2x-35$$

Add 2x to both sides.

$$7x^{2}+19x+17+2x=-35$$

Combine 19x and 2x to get 21x.

$$7x^{2}+21x+17=-35$$

Add 35 to both sides.

$$7x^{2}+21x+17+35=0$$

Add 17 and 35 to get 52.

$$7x^{2}+21x+52=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 21 for b, and 52 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-21±\sqrt{21^{2}-4\times 7\times 52}}{2\times 7}$$

Square 21.

$$x=\frac{-21±\sqrt{441-4\times 7\times 52}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-21±\sqrt{441-28\times 52}}{2\times 7}$$

Multiply -28 times 52.

$$x=\frac{-21±\sqrt{441-1456}}{2\times 7}$$

Add 441 to -1456.

$$x=\frac{-21±\sqrt{-1015}}{2\times 7}$$

Take the square root of -1015.

$$x=\frac{-21±\sqrt{1015}i}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-21±\sqrt{1015}i}{14}$$

Now solve the equation x=\frac{-21±\sqrt{1015}i}{14} when ± is plus. Add -21 to i\sqrt{1015}.

$$x=\frac{-21+\sqrt{1015}i}{14}$$

Divide -21+i\sqrt{1015} by 14.

$$x=\frac{\sqrt{1015}i}{14}-\frac{3}{2}$$

Now solve the equation x=\frac{-21±\sqrt{1015}i}{14} when ± is minus. Subtract i\sqrt{1015} from -21.

$$x=\frac{-\sqrt{1015}i-21}{14}$$

Divide -21-i\sqrt{1015} by 14.

$$x=-\frac{\sqrt{1015}i}{14}-\frac{3}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{1015}i}{14}-\frac{3}{2} x=-\frac{\sqrt{1015}i}{14}-\frac{3}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{2}{3},1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(3x+2\right), the least common multiple of x-1,3x+2,3x^{2}-x-2.

$$\left(3x+2\right)\left(x+7\right)+\left(x-1\right)\left(x-3\right)=\left(x-1\right)\left(3x+2\right)\times 2-\left(9x^{2}+31\right)$$

Use the distributive property to multiply 3x+2 by x+7 and combine like terms.

$$3x^{2}+23x+14+\left(x-1\right)\left(x-3\right)=\left(x-1\right)\left(3x+2\right)\times 2-\left(9x^{2}+31\right)$$

Use the distributive property to multiply x-1 by x-3 and combine like terms.

$$3x^{2}+23x+14+x^{2}-4x+3=\left(x-1\right)\left(3x+2\right)\times 2-\left(9x^{2}+31\right)$$

Combine 3x^{2} and x^{2} to get 4x^{2}.

$$4x^{2}+23x+14-4x+3=\left(x-1\right)\left(3x+2\right)\times 2-\left(9x^{2}+31\right)$$

Combine 23x and -4x to get 19x.

$$4x^{2}+19x+14+3=\left(x-1\right)\left(3x+2\right)\times 2-\left(9x^{2}+31\right)$$

Add 14 and 3 to get 17.

$$4x^{2}+19x+17=\left(x-1\right)\left(3x+2\right)\times 2-\left(9x^{2}+31\right)$$

Use the distributive property to multiply x-1 by 3x+2 and combine like terms.

$$4x^{2}+19x+17=\left(3x^{2}-x-2\right)\times 2-\left(9x^{2}+31\right)$$

Use the distributive property to multiply 3x^{2}-x-2 by 2.

$$4x^{2}+19x+17=6x^{2}-2x-4-\left(9x^{2}+31\right)$$

To find the opposite of 9x^{2}+31, find the opposite of each term.

$$4x^{2}+19x+17=6x^{2}-2x-4-9x^{2}-31$$

Combine 6x^{2} and -9x^{2} to get -3x^{2}.

$$4x^{2}+19x+17=-3x^{2}-2x-4-31$$

Subtract 31 from -4 to get -35.

$$4x^{2}+19x+17=-3x^{2}-2x-35$$

Add 3x^{2} to both sides.

$$4x^{2}+19x+17+3x^{2}=-2x-35$$

Combine 4x^{2} and 3x^{2} to get 7x^{2}.

$$7x^{2}+19x+17=-2x-35$$

Add 2x to both sides.

$$7x^{2}+19x+17+2x=-35$$

Combine 19x and 2x to get 21x.

$$7x^{2}+21x+17=-35$$

Subtract 17 from both sides.

$$7x^{2}+21x=-35-17$$

Subtract 17 from -35 to get -52.

$$7x^{2}+21x=-52$$

Divide both sides by 7.

$$\frac{7x^{2}+21x}{7}=-\frac{52}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}+\frac{21}{7}x=-\frac{52}{7}$$

Divide 21 by 7.

$$x^{2}+3x=-\frac{52}{7}$$

Divide 3, the coefficient of the x term, by 2 to get \frac{3}{2}. Then add the square of \frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=-\frac{52}{7}+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=-\frac{52}{7}+\frac{9}{4}$$

Add -\frac{52}{7} to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+3x+\frac{9}{4}=-\frac{145}{28}$$

Factor x^{2}+3x+\frac{9}{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}.

$$\left(x+\frac{3}{2}\right)^{2}=-\frac{145}{28}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{-\frac{145}{28}}$$

Simplify.

$$x+\frac{3}{2}=\frac{\sqrt{1015}i}{14} x+\frac{3}{2}=-\frac{\sqrt{1015}i}{14}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=\frac{\sqrt{1015}i}{14}-\frac{3}{2} x=-\frac{\sqrt{1015}i}{14}-\frac{3}{2}$$