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

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

Solve for x

$$x=-\frac{1}{6}\approx -0.166666667$$

$$x=1$$

Steps Using Factoring By Grouping

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

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

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

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

Use the distributive property to multiply 2 by x-1.

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

Combine 3x and 2x to get 5x.

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

Subtract 2 from 3 to get 1.

$$5x+1=6x^{2}$$

Subtract 6x^{2} from both sides.

$$5x+1-6x^{2}=0$$

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

$$-6x^{2}+5x+1=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -6x^{2}+ax+bx+1. To find a and b, set up a system to be solved.

$$a+b=5 ab=-6=-6$$

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 -6.

$$-1,6 -2,3$$

Calculate the sum for each pair.

$$-1+6=5 -2+3=1$$

The solution is the pair that gives sum 5.

$$a=6 b=-1$$

Rewrite -6x^{2}+5x+1 as \left(-6x^{2}+6x\right)+\left(-x+1\right).

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

Factor out 6x in -6x^{2}+6x.

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

Factor out common term -x+1 by using distributive property.

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

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

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

Steps Using the Quadratic Formula

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

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

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

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

Use the distributive property to multiply 2 by x-1.

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

Combine 3x and 2x to get 5x.

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

Subtract 2 from 3 to get 1.

$$5x+1=6x^{2}$$

Subtract 6x^{2} from both sides.

$$5x+1-6x^{2}=0$$

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.

$$-6x^{2}+5x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 5 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square 5.

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

Multiply -4 times -6.

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

Add 25 to 24.

$$x=\frac{-5±\sqrt{49}}{2\left(-6\right)}$$

Take the square root of 49.

$$x=\frac{-5±7}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-5±7}{-12}$$

Now solve the equation x=\frac{-5±7}{-12} when ± is plus. Add -5 to 7.

$$x=\frac{2}{-12}$$

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

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

Now solve the equation x=\frac{-5±7}{-12} when ± is minus. Subtract 7 from -5.

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

Divide -12 by -12.

$$x=1$$

The equation is now solved.

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

Steps for Completing the Square

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

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

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

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

Use the distributive property to multiply 2 by x-1.

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

Combine 3x and 2x to get 5x.

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

Subtract 2 from 3 to get 1.

$$5x+1=6x^{2}$$

Subtract 6x^{2} from both sides.

$$5x+1-6x^{2}=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$5x-6x^{2}=-1$$

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.

$$-6x^{2}+5x=-1$$

Divide both sides by -6.

$$\frac{-6x^{2}+5x}{-6}=-\frac{1}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\frac{5}{-6}x=-\frac{1}{-6}$$

Divide 5 by -6.

$$x^{2}-\frac{5}{6}x=-\frac{1}{-6}$$

Divide -1 by -6.

$$x^{2}-\frac{5}{6}x=\frac{1}{6}$$

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.

$$x^{2}-\frac{5}{6}x+\left(-\frac{5}{12}\right)^{2}=\frac{1}{6}+\left(-\frac{5}{12}\right)^{2}$$

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

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=\frac{1}{6}+\frac{25}{144}$$

Add \frac{1}{6} to \frac{25}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=\frac{49}{144}$$

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}.

$$\left(x-\frac{5}{12}\right)^{2}=\frac{49}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{12}\right)^{2}}=\sqrt{\frac{49}{144}}$$

Simplify.

$$x-\frac{5}{12}=\frac{7}{12} x-\frac{5}{12}=-\frac{7}{12}$$

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

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