Solve 2/5left(x+2right)+2/15x=1/30

$$\frac{ 2 }{ 5 \left( x+2 \right) } + \frac{ 2 }{ 15x } = \frac{ 1 }{ 30 }$$

Solve for x

$$x=\sqrt{57}+7\approx 14.549834435$$

$$x=7-\sqrt{57}\approx -0.549834435$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 30x\left(x+2\right), the least common multiple of 5\left(x+2\right),15x,30.

$$6x\times 2+\left(2x+4\right)\times 2=x\left(x+2\right)$$

Multiply 6 and 2 to get 12.

$$12x+\left(2x+4\right)\times 2=x\left(x+2\right)$$

Use the distributive property to multiply 2x+4 by 2.

$$12x+4x+8=x\left(x+2\right)$$

Combine 12x and 4x to get 16x.

$$16x+8=x\left(x+2\right)$$

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

$$16x+8=x^{2}+2x$$

Subtract x^{2} from both sides.

$$16x+8-x^{2}=2x$$

Subtract 2x from both sides.

$$16x+8-x^{2}-2x=0$$

Combine 16x and -2x to get 14x.

$$14x+8-x^{2}=0$$

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.

$$-x^{2}+14x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 14 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-14±\sqrt{196+4\times 8}}{2\left(-1\right)}$$

Multiply 4 times 8.

$$x=\frac{-14±\sqrt{196+32}}{2\left(-1\right)}$$

Add 196 to 32.

$$x=\frac{-14±\sqrt{228}}{2\left(-1\right)}$$

Take the square root of 228.

$$x=\frac{-14±2\sqrt{57}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-14±2\sqrt{57}}{-2}$$

Now solve the equation x=\frac{-14±2\sqrt{57}}{-2} when ± is plus. Add -14 to 2\sqrt{57}.

$$x=\frac{2\sqrt{57}-14}{-2}$$

Divide -14+2\sqrt{57} by -2.

$$x=7-\sqrt{57}$$

Now solve the equation x=\frac{-14±2\sqrt{57}}{-2} when ± is minus. Subtract 2\sqrt{57} from -14.

$$x=\frac{-2\sqrt{57}-14}{-2}$$

Divide -14-2\sqrt{57} by -2.

$$x=\sqrt{57}+7$$

The equation is now solved.

$$x=7-\sqrt{57} x=\sqrt{57}+7$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 30x\left(x+2\right), the least common multiple of 5\left(x+2\right),15x,30.

$$6x\times 2+\left(2x+4\right)\times 2=x\left(x+2\right)$$

Multiply 6 and 2 to get 12.

$$12x+\left(2x+4\right)\times 2=x\left(x+2\right)$$

Use the distributive property to multiply 2x+4 by 2.

$$12x+4x+8=x\left(x+2\right)$$

Combine 12x and 4x to get 16x.

$$16x+8=x\left(x+2\right)$$

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

$$16x+8=x^{2}+2x$$

Subtract x^{2} from both sides.

$$16x+8-x^{2}=2x$$

Subtract 2x from both sides.

$$16x+8-x^{2}-2x=0$$

Combine 16x and -2x to get 14x.

$$14x+8-x^{2}=0$$

Subtract 8 from both sides. Anything subtracted from zero gives its negation.

$$14x-x^{2}=-8$$

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.

$$-x^{2}+14x=-8$$

Divide both sides by -1.

$$\frac{-x^{2}+14x}{-1}=-\frac{8}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{14}{-1}x=-\frac{8}{-1}$$

Divide 14 by -1.

$$x^{2}-14x=-\frac{8}{-1}$$

Divide -8 by -1.

$$x^{2}-14x=8$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=8+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=8+49$$

Add 8 to 49.

$$x^{2}-14x+49=57$$

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=57$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{57}$$

Simplify.

$$x-7=\sqrt{57} x-7=-\sqrt{57}$$

Add 7 to both sides of the equation.

$$x=\sqrt{57}+7 x=7-\sqrt{57}$$