Solve x^2-1/3-x-1/6=2+x/9

$$\frac { x ^ { 2 } - 1 } { 3 } - \frac { x - 1 } { 6 } = 2 + \frac { x } { 9 }$$

Solve for x

$$x = -\frac{13}{6} = -2\frac{1}{6} \approx -2.166666667$$

$$x=3$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 18, the least common multiple of 3,6,9.

$$6\left(x^{2}-1\right)-3\left(x-1\right)=36+2x$$

Use the distributive property to multiply 6 by x^{2}-1.

$$6x^{2}-6-3\left(x-1\right)=36+2x$$

Use the distributive property to multiply -3 by x-1.

$$6x^{2}-6-3x+3=36+2x$$

Add -6 and 3 to get -3.

$$6x^{2}-3-3x=36+2x$$

Subtract 36 from both sides.

$$6x^{2}-3-3x-36=2x$$

Subtract 36 from -3 to get -39.

$$6x^{2}-39-3x=2x$$

Subtract 2x from both sides.

$$6x^{2}-39-3x-2x=0$$

Combine -3x and -2x to get -5x.

$$6x^{2}-39-5x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$6x^{2}-5x-39=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 6x^{2}+ax+bx-39. To find a and b, set up a system to be solved.

$$a+b=-5 ab=6\left(-39\right)=-234$$

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 -234.

$$1,-234 2,-117 3,-78 6,-39 9,-26 13,-18$$

Calculate the sum for each pair.

$$1-234=-233 2-117=-115 3-78=-75 6-39=-33 9-26=-17 13-18=-5$$

The solution is the pair that gives sum -5.

$$a=-18 b=13$$

Rewrite 6x^{2}-5x-39 as \left(6x^{2}-18x\right)+\left(13x-39\right).

$$\left(6x^{2}-18x\right)+\left(13x-39\right)$$

Factor out 6x in the first and 13 in the second group.

$$6x\left(x-3\right)+13\left(x-3\right)$$

Factor out common term x-3 by using distributive property.

$$\left(x-3\right)\left(6x+13\right)$$

To find equation solutions, solve x-3=0 and 6x+13=0.

$$x=3 x=-\frac{13}{6}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 18, the least common multiple of 3,6,9.

$$6\left(x^{2}-1\right)-3\left(x-1\right)=36+2x$$

Use the distributive property to multiply 6 by x^{2}-1.

$$6x^{2}-6-3\left(x-1\right)=36+2x$$

Use the distributive property to multiply -3 by x-1.

$$6x^{2}-6-3x+3=36+2x$$

Add -6 and 3 to get -3.

$$6x^{2}-3-3x=36+2x$$

Subtract 36 from both sides.

$$6x^{2}-3-3x-36=2x$$

Subtract 36 from -3 to get -39.

$$6x^{2}-39-3x=2x$$

Subtract 2x from both sides.

$$6x^{2}-39-3x-2x=0$$

Combine -3x and -2x to get -5x.

$$6x^{2}-39-5x=0$$

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.

$$6x^{2}-5x-39=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -5 for b, and -39 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 6\left(-39\right)}}{2\times 6}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 6\left(-39\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-5\right)±\sqrt{25-24\left(-39\right)}}{2\times 6}$$

Multiply -24 times -39.

$$x=\frac{-\left(-5\right)±\sqrt{25+936}}{2\times 6}$$

Add 25 to 936.

$$x=\frac{-\left(-5\right)±\sqrt{961}}{2\times 6}$$

Take the square root of 961.

$$x=\frac{-\left(-5\right)±31}{2\times 6}$$

The opposite of -5 is 5.

$$x=\frac{5±31}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{5±31}{12}$$

Now solve the equation x=\frac{5±31}{12} when ± is plus. Add 5 to 31.

$$x=\frac{36}{12}$$

Divide 36 by 12.

$$x=3$$

Now solve the equation x=\frac{5±31}{12} when ± is minus. Subtract 31 from 5.

$$x=-\frac{26}{12}$$

Reduce the fraction \frac{-26}{12} to lowest terms by extracting and canceling out 2.

$$x=-\frac{13}{6}$$

The equation is now solved.

$$x=3 x=-\frac{13}{6}$$

Steps for Completing the Square

Multiply both sides of the equation by 18, the least common multiple of 3,6,9.

$$6\left(x^{2}-1\right)-3\left(x-1\right)=36+2x$$

Use the distributive property to multiply 6 by x^{2}-1.

$$6x^{2}-6-3\left(x-1\right)=36+2x$$

Use the distributive property to multiply -3 by x-1.

$$6x^{2}-6-3x+3=36+2x$$

Add -6 and 3 to get -3.

$$6x^{2}-3-3x=36+2x$$

Subtract 2x from both sides.

$$6x^{2}-3-3x-2x=36$$

Combine -3x and -2x to get -5x.

$$6x^{2}-3-5x=36$$

Add 3 to both sides.

$$6x^{2}-5x=36+3$$

Add 36 and 3 to get 39.

$$6x^{2}-5x=39$$

Divide both sides by 6.

$$\frac{6x^{2}-5x}{6}=\frac{39}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}-\frac{5}{6}x=\frac{39}{6}$$

Reduce the fraction \frac{39}{6} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{5}{6}x=\frac{13}{2}$$

Divide -\frac{5}{6}, the coefficient of the x term, by 2 to get -\frac{5}{12}. Then add the square of -\frac{5}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{6}x+\left(-\frac{5}{12}\right)^{2}=\frac{13}{2}+\left(-\frac{5}{12}\right)^{2}$$

Square -\frac{5}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=\frac{13}{2}+\frac{25}{144}$$

Add \frac{13}{2} to \frac{25}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=\frac{961}{144}$$

Factor x^{2}-\frac{5}{6}x+\frac{25}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{12}\right)^{2}=\frac{961}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{12}\right)^{2}}=\sqrt{\frac{961}{144}}$$

Simplify.

$$x-\frac{5}{12}=\frac{31}{12} x-\frac{5}{12}=-\frac{31}{12}$$

Add \frac{5}{12} to both sides of the equation.

$$x=3 x=-\frac{13}{6}$$