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

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

Solve for x

$$x=-1$$

$$x = \frac{19}{6} = 3\frac{1}{6} \approx 3.166666667$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 6, the least common multiple of 2,3.

$$6x+3\left(3x+1\right)-2\left(x-2\right)=6x^{2}-12$$

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

$$6x+9x+3-2\left(x-2\right)=6x^{2}-12$$

Combine 6x and 9x to get 15x.

$$15x+3-2\left(x-2\right)=6x^{2}-12$$

Use the distributive property to multiply -2 by x-2.

$$15x+3-2x+4=6x^{2}-12$$

Combine 15x and -2x to get 13x.

$$13x+3+4=6x^{2}-12$$

Add 3 and 4 to get 7.

$$13x+7=6x^{2}-12$$

Subtract 6x^{2} from both sides.

$$13x+7-6x^{2}=-12$$

Add 12 to both sides.

$$13x+7-6x^{2}+12=0$$

Add 7 and 12 to get 19.

$$13x+19-6x^{2}=0$$

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

$$-6x^{2}+13x+19=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -6x^{2}+ax+bx+19. To find a and b, set up a system to be solved.

$$a+b=13 ab=-6\times 19=-114$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -114.

$$-1,114 -2,57 -3,38 -6,19$$

Calculate the sum for each pair.

$$-1+114=113 -2+57=55 -3+38=35 -6+19=13$$

The solution is the pair that gives sum 13.

$$a=19 b=-6$$

Rewrite -6x^{2}+13x+19 as \left(-6x^{2}+19x\right)+\left(-6x+19\right).

$$\left(-6x^{2}+19x\right)+\left(-6x+19\right)$$

Factor out -x in the first and -1 in the second group.

$$-x\left(6x-19\right)-\left(6x-19\right)$$

Factor out common term 6x-19 by using distributive property.

$$\left(6x-19\right)\left(-x-1\right)$$

To find equation solutions, solve 6x-19=0 and -x-1=0.

$$x=\frac{19}{6} x=-1$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 6, the least common multiple of 2,3.

$$6x+3\left(3x+1\right)-2\left(x-2\right)=6x^{2}-12$$

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

$$6x+9x+3-2\left(x-2\right)=6x^{2}-12$$

Combine 6x and 9x to get 15x.

$$15x+3-2\left(x-2\right)=6x^{2}-12$$

Use the distributive property to multiply -2 by x-2.

$$15x+3-2x+4=6x^{2}-12$$

Combine 15x and -2x to get 13x.

$$13x+3+4=6x^{2}-12$$

Add 3 and 4 to get 7.

$$13x+7=6x^{2}-12$$

Subtract 6x^{2} from both sides.

$$13x+7-6x^{2}=-12$$

Add 12 to both sides.

$$13x+7-6x^{2}+12=0$$

Add 7 and 12 to get 19.

$$13x+19-6x^{2}=0$$

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.

$$-6x^{2}+13x+19=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 13 for b, and 19 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-6\right)\times 19}}{2\left(-6\right)}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\left(-6\right)\times 19}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-13±\sqrt{169+24\times 19}}{2\left(-6\right)}$$

Multiply 24 times 19.

$$x=\frac{-13±\sqrt{169+456}}{2\left(-6\right)}$$

Add 169 to 456.

$$x=\frac{-13±\sqrt{625}}{2\left(-6\right)}$$

Take the square root of 625.

$$x=\frac{-13±25}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-13±25}{-12}$$

Now solve the equation x=\frac{-13±25}{-12} when ± is plus. Add -13 to 25.

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

Divide 12 by -12.

$$x=-1$$

Now solve the equation x=\frac{-13±25}{-12} when ± is minus. Subtract 25 from -13.

$$x=-\frac{38}{-12}$$

Reduce the fraction \frac{-38}{-12} to lowest terms by extracting and canceling out 2.

$$x=\frac{19}{6}$$

The equation is now solved.

$$x=-1 x=\frac{19}{6}$$

Steps for Completing the Square

Multiply both sides of the equation by 6, the least common multiple of 2,3.

$$6x+3\left(3x+1\right)-2\left(x-2\right)=6x^{2}-12$$

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

$$6x+9x+3-2\left(x-2\right)=6x^{2}-12$$

Combine 6x and 9x to get 15x.

$$15x+3-2\left(x-2\right)=6x^{2}-12$$

Use the distributive property to multiply -2 by x-2.

$$15x+3-2x+4=6x^{2}-12$$

Combine 15x and -2x to get 13x.

$$13x+3+4=6x^{2}-12$$

Add 3 and 4 to get 7.

$$13x+7=6x^{2}-12$$

Subtract 6x^{2} from both sides.

$$13x+7-6x^{2}=-12$$

Subtract 7 from both sides.

$$13x-6x^{2}=-12-7$$

Subtract 7 from -12 to get -19.

$$13x-6x^{2}=-19$$

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.

$$-6x^{2}+13x=-19$$

Divide both sides by -6.

$$\frac{-6x^{2}+13x}{-6}=-\frac{19}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\frac{13}{-6}x=-\frac{19}{-6}$$

Divide 13 by -6.

$$x^{2}-\frac{13}{6}x=-\frac{19}{-6}$$

Divide -19 by -6.

$$x^{2}-\frac{13}{6}x=\frac{19}{6}$$

Divide -\frac{13}{6}, the coefficient of the x term, by 2 to get -\frac{13}{12}. Then add the square of -\frac{13}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{6}x+\left(-\frac{13}{12}\right)^{2}=\frac{19}{6}+\left(-\frac{13}{12}\right)^{2}$$

Square -\frac{13}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=\frac{19}{6}+\frac{169}{144}$$

Add \frac{19}{6} to \frac{169}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=\frac{625}{144}$$

Factor x^{2}-\frac{13}{6}x+\frac{169}{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}.

$$\left(x-\frac{13}{12}\right)^{2}=\frac{625}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{12}\right)^{2}}=\sqrt{\frac{625}{144}}$$

Simplify.

$$x-\frac{13}{12}=\frac{25}{12} x-\frac{13}{12}=-\frac{25}{12}$$

Add \frac{13}{12} to both sides of the equation.

$$x=\frac{19}{6} x=-1$$