Solve 2=3x-30x/15+x

$$2 = 3 x - \frac { 30 x } { 15 + x }$$

Solve for x

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

$$x=-6$$

Steps Using Factoring By Grouping

Variable x cannot be equal to -15 since division by zero is not defined. Multiply both sides of the equation by x+15.

$$2\left(x+15\right)=3x\left(x+15\right)-30x$$

Use the distributive property to multiply 2 by x+15.

$$2x+30=3x\left(x+15\right)-30x$$

Use the distributive property to multiply 3x by x+15.

$$2x+30=3x^{2}+45x-30x$$

Combine 45x and -30x to get 15x.

$$2x+30=3x^{2}+15x$$

Subtract 3x^{2} from both sides.

$$2x+30-3x^{2}=15x$$

Subtract 15x from both sides.

$$2x+30-3x^{2}-15x=0$$

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

$$-13x+30-3x^{2}=0$$

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

$$-3x^{2}-13x+30=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -3x^{2}+ax+bx+30. To find a and b, set up a system to be solved.

$$a+b=-13 ab=-3\times 30=-90$$

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 -90.

$$1,-90 2,-45 3,-30 5,-18 6,-15 9,-10$$

Calculate the sum for each pair.

$$1-90=-89 2-45=-43 3-30=-27 5-18=-13 6-15=-9 9-10=-1$$

The solution is the pair that gives sum -13.

$$a=5 b=-18$$

Rewrite -3x^{2}-13x+30 as \left(-3x^{2}+5x\right)+\left(-18x+30\right).

$$\left(-3x^{2}+5x\right)+\left(-18x+30\right)$$

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

$$-x\left(3x-5\right)-6\left(3x-5\right)$$

Factor out common term 3x-5 by using distributive property.

$$\left(3x-5\right)\left(-x-6\right)$$

To find equation solutions, solve 3x-5=0 and -x-6=0.

$$x=\frac{5}{3} x=-6$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -15 since division by zero is not defined. Multiply both sides of the equation by x+15.

$$2\left(x+15\right)=3x\left(x+15\right)-30x$$

Use the distributive property to multiply 2 by x+15.

$$2x+30=3x\left(x+15\right)-30x$$

Use the distributive property to multiply 3x by x+15.

$$2x+30=3x^{2}+45x-30x$$

Combine 45x and -30x to get 15x.

$$2x+30=3x^{2}+15x$$

Subtract 3x^{2} from both sides.

$$2x+30-3x^{2}=15x$$

Subtract 15x from both sides.

$$2x+30-3x^{2}-15x=0$$

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

$$-13x+30-3x^{2}=0$$

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.

$$-3x^{2}-13x+30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -13 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\left(-3\right)\times 30}}{2\left(-3\right)}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\left(-3\right)\times 30}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-13\right)±\sqrt{169+12\times 30}}{2\left(-3\right)}$$

Multiply 12 times 30.

$$x=\frac{-\left(-13\right)±\sqrt{169+360}}{2\left(-3\right)}$$

Add 169 to 360.

$$x=\frac{-\left(-13\right)±\sqrt{529}}{2\left(-3\right)}$$

Take the square root of 529.

$$x=\frac{-\left(-13\right)±23}{2\left(-3\right)}$$

The opposite of -13 is 13.

$$x=\frac{13±23}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{13±23}{-6}$$

Now solve the equation x=\frac{13±23}{-6} when ± is plus. Add 13 to 23.

$$x=\frac{36}{-6}$$

Divide 36 by -6.

$$x=-6$$

Now solve the equation x=\frac{13±23}{-6} when ± is minus. Subtract 23 from 13.

$$x=-\frac{10}{-6}$$

Reduce the fraction \frac{-10}{-6} to lowest terms by extracting and canceling out 2.

$$x=\frac{5}{3}$$

The equation is now solved.

$$x=-6 x=\frac{5}{3}$$

Steps for Completing the Square

Variable x cannot be equal to -15 since division by zero is not defined. Multiply both sides of the equation by x+15.

$$2\left(x+15\right)=3x\left(x+15\right)-30x$$

Use the distributive property to multiply 2 by x+15.

$$2x+30=3x\left(x+15\right)-30x$$

Use the distributive property to multiply 3x by x+15.

$$2x+30=3x^{2}+45x-30x$$

Combine 45x and -30x to get 15x.

$$2x+30=3x^{2}+15x$$

Subtract 3x^{2} from both sides.

$$2x+30-3x^{2}=15x$$

Subtract 15x from both sides.

$$2x+30-3x^{2}-15x=0$$

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

$$-13x+30-3x^{2}=0$$

Subtract 30 from both sides. Anything subtracted from zero gives its negation.

$$-13x-3x^{2}=-30$$

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.

$$-3x^{2}-13x=-30$$

Divide both sides by -3.

$$\frac{-3x^{2}-13x}{-3}=-\frac{30}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{13}{-3}\right)x=-\frac{30}{-3}$$

Divide -13 by -3.

$$x^{2}+\frac{13}{3}x=-\frac{30}{-3}$$

Divide -30 by -3.

$$x^{2}+\frac{13}{3}x=10$$

Divide \frac{13}{3}, the coefficient of the x term, by 2 to get \frac{13}{6}. Then add the square of \frac{13}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{13}{3}x+\left(\frac{13}{6}\right)^{2}=10+\left(\frac{13}{6}\right)^{2}$$

Square \frac{13}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{13}{3}x+\frac{169}{36}=10+\frac{169}{36}$$

Add 10 to \frac{169}{36}.

$$x^{2}+\frac{13}{3}x+\frac{169}{36}=\frac{529}{36}$$

Factor x^{2}+\frac{13}{3}x+\frac{169}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{13}{6}\right)^{2}=\frac{529}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{13}{6}\right)^{2}}=\sqrt{\frac{529}{36}}$$

Simplify.

$$x+\frac{13}{6}=\frac{23}{6} x+\frac{13}{6}=-\frac{23}{6}$$

Subtract \frac{13}{6} from both sides of the equation.

$$x=\frac{5}{3} x=-6$$