Solve 7/x+2+10+2x=1

$$\frac{ 7 }{ x+2 } +10+2x=1$$

Solve for x (complex solution)

$$x=\frac{-13+\sqrt{31}i}{4}\approx -3.25+1.391941091i$$

$$x=\frac{-\sqrt{31}i-13}{4}\approx -3.25-1.391941091i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by x+2.

$$7+\left(x+2\right)\times 10+2x\left(x+2\right)=x+2$$

Use the distributive property to multiply x+2 by 10.

$$7+10x+20+2x\left(x+2\right)=x+2$$

Add 7 and 20 to get 27.

$$27+10x+2x\left(x+2\right)=x+2$$

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

$$27+10x+2x^{2}+4x=x+2$$

Combine 10x and 4x to get 14x.

$$27+14x+2x^{2}=x+2$$

Subtract x from both sides.

$$27+14x+2x^{2}-x=2$$

Combine 14x and -x to get 13x.

$$27+13x+2x^{2}=2$$

Subtract 2 from both sides.

$$27+13x+2x^{2}-2=0$$

Subtract 2 from 27 to get 25.

$$25+13x+2x^{2}=0$$

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.

$$2x^{2}+13x+25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 13 for b, and 25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\times 2\times 25}}{2\times 2}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\times 2\times 25}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-13±\sqrt{169-8\times 25}}{2\times 2}$$

Multiply -8 times 25.

$$x=\frac{-13±\sqrt{169-200}}{2\times 2}$$

Add 169 to -200.

$$x=\frac{-13±\sqrt{-31}}{2\times 2}$$

Take the square root of -31.

$$x=\frac{-13±\sqrt{31}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-13±\sqrt{31}i}{4}$$

Now solve the equation x=\frac{-13±\sqrt{31}i}{4} when ± is plus. Add -13 to i\sqrt{31}.

$$x=\frac{-13+\sqrt{31}i}{4}$$

Now solve the equation x=\frac{-13±\sqrt{31}i}{4} when ± is minus. Subtract i\sqrt{31} from -13.

$$x=\frac{-\sqrt{31}i-13}{4}$$

The equation is now solved.

$$x=\frac{-13+\sqrt{31}i}{4} x=\frac{-\sqrt{31}i-13}{4}$$

Steps for Completing the Square

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by x+2.

$$7+\left(x+2\right)\times 10+2x\left(x+2\right)=x+2$$

Use the distributive property to multiply x+2 by 10.

$$7+10x+20+2x\left(x+2\right)=x+2$$

Add 7 and 20 to get 27.

$$27+10x+2x\left(x+2\right)=x+2$$

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

$$27+10x+2x^{2}+4x=x+2$$

Combine 10x and 4x to get 14x.

$$27+14x+2x^{2}=x+2$$

Subtract x from both sides.

$$27+14x+2x^{2}-x=2$$

Combine 14x and -x to get 13x.

$$27+13x+2x^{2}=2$$

Subtract 27 from both sides.

$$13x+2x^{2}=2-27$$

Subtract 27 from 2 to get -25.

$$13x+2x^{2}=-25$$

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.

$$2x^{2}+13x=-25$$

Divide both sides by 2.

$$\frac{2x^{2}+13x}{2}=-\frac{25}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{13}{2}x=-\frac{25}{2}$$

Divide \frac{13}{2}, the coefficient of the x term, by 2 to get \frac{13}{4}. Then add the square of \frac{13}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{13}{2}x+\left(\frac{13}{4}\right)^{2}=-\frac{25}{2}+\left(\frac{13}{4}\right)^{2}$$

Square \frac{13}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{13}{2}x+\frac{169}{16}=-\frac{25}{2}+\frac{169}{16}$$

Add -\frac{25}{2} to \frac{169}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{13}{2}x+\frac{169}{16}=-\frac{31}{16}$$

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

$$\left(x+\frac{13}{4}\right)^{2}=-\frac{31}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{13}{4}\right)^{2}}=\sqrt{-\frac{31}{16}}$$

Simplify.

$$x+\frac{13}{4}=\frac{\sqrt{31}i}{4} x+\frac{13}{4}=-\frac{\sqrt{31}i}{4}$$

Subtract \frac{13}{4} from both sides of the equation.

$$x=\frac{-13+\sqrt{31}i}{4} x=\frac{-\sqrt{31}i-13}{4}$$