Solve x+2/x=x+1/32

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

Solve for x

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

$$x=2$$

Steps Using Factoring By Grouping

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3x, the least common multiple of x,3.

$$3\left(x+2\right)=x\left(x+1\right)\times 2$$

Use the distributive property to multiply 3 by x+2.

$$3x+6=x\left(x+1\right)\times 2$$

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

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

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

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

Subtract 2x^{2} from both sides.

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

Subtract 2x from both sides.

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

Combine 3x and -2x to get x.

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

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

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

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

$$a+b=1 ab=-2\times 6=-12$$

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 -12.

$$-1,12 -2,6 -3,4$$

Calculate the sum for each pair.

$$-1+12=11 -2+6=4 -3+4=1$$

The solution is the pair that gives sum 1.

$$a=4 b=-3$$

Rewrite -2x^{2}+x+6 as \left(-2x^{2}+4x\right)+\left(-3x+6\right).

$$\left(-2x^{2}+4x\right)+\left(-3x+6\right)$$

Factor out 2x in the first and 3 in the second group.

$$2x\left(-x+2\right)+3\left(-x+2\right)$$

Factor out common term -x+2 by using distributive property.

$$\left(-x+2\right)\left(2x+3\right)$$

To find equation solutions, solve -x+2=0 and 2x+3=0.

$$x=2 x=-\frac{3}{2}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3x, the least common multiple of x,3.

$$3\left(x+2\right)=x\left(x+1\right)\times 2$$

Use the distributive property to multiply 3 by x+2.

$$3x+6=x\left(x+1\right)\times 2$$

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

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

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

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

Subtract 2x^{2} from both sides.

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

Subtract 2x from both sides.

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

Combine 3x and -2x to get x.

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

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.

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

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

$$x=\frac{-1±\sqrt{1^{2}-4\left(-2\right)\times 6}}{2\left(-2\right)}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-2\right)\times 6}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-1±\sqrt{1+8\times 6}}{2\left(-2\right)}$$

Multiply 8 times 6.

$$x=\frac{-1±\sqrt{1+48}}{2\left(-2\right)}$$

Add 1 to 48.

$$x=\frac{-1±\sqrt{49}}{2\left(-2\right)}$$

Take the square root of 49.

$$x=\frac{-1±7}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-1±7}{-4}$$

Now solve the equation x=\frac{-1±7}{-4} when ± is plus. Add -1 to 7.

$$x=\frac{6}{-4}$$

Reduce the fraction \frac{6}{-4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{3}{2}$$

Now solve the equation x=\frac{-1±7}{-4} when ± is minus. Subtract 7 from -1.

$$x=-\frac{8}{-4}$$

Divide -8 by -4.

$$x=2$$

The equation is now solved.

$$x=-\frac{3}{2} x=2$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3x, the least common multiple of x,3.

$$3\left(x+2\right)=x\left(x+1\right)\times 2$$

Use the distributive property to multiply 3 by x+2.

$$3x+6=x\left(x+1\right)\times 2$$

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

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

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

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

Subtract 2x^{2} from both sides.

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

Subtract 2x from both sides.

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

Combine 3x and -2x to get x.

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

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

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

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.

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

Divide both sides by -2.

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

Dividing by -2 undoes the multiplication by -2.

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

Divide 1 by -2.

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

Divide -6 by -2.

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

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=3+\left(-\frac{1}{4}\right)^{2}$$

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=3+\frac{1}{16}$$

Add 3 to \frac{1}{16}.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{49}{16}$$

Factor x^{2}-\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{49}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Simplify.

$$x-\frac{1}{4}=\frac{7}{4} x-\frac{1}{4}=-\frac{7}{4}$$

Add \frac{1}{4} to both sides of the equation.

$$x=2 x=-\frac{3}{2}$$