Solve (x+2)^2/7x+2x/5=0

$$\frac { ( x + 2 ) ^ { 2 } } { 7 x } + \frac { 2 x } { 5 } = 0$$

Solve for x (complex solution)

$$x=\frac{-10+2\sqrt{70}i}{19}\approx -0.526315789+0.880694765i$$

$$x=\frac{-2\sqrt{70}i-10}{19}\approx -0.526315789-0.880694765i$$

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 35x, the least common multiple of 7x,5.

$$5\left(x+2\right)^{2}+7x\times 2x=0$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2\right)^{2}.

$$5\left(x^{2}+4x+4\right)+7x\times 2x=0$$

Use the distributive property to multiply 5 by x^{2}+4x+4.

$$5x^{2}+20x+20+7x\times 2x=0$$

Multiply x and x to get x^{2}.

$$5x^{2}+20x+20+7x^{2}\times 2=0$$

Multiply 7 and 2 to get 14.

$$5x^{2}+20x+20+14x^{2}=0$$

Combine 5x^{2} and 14x^{2} to get 19x^{2}.

$$19x^{2}+20x+20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 19 for a, 20 for b, and 20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\times 19\times 20}}{2\times 19}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\times 19\times 20}}{2\times 19}$$

Multiply -4 times 19.

$$x=\frac{-20±\sqrt{400-76\times 20}}{2\times 19}$$

Multiply -76 times 20.

$$x=\frac{-20±\sqrt{400-1520}}{2\times 19}$$

Add 400 to -1520.

$$x=\frac{-20±\sqrt{-1120}}{2\times 19}$$

Take the square root of -1120.

$$x=\frac{-20±4\sqrt{70}i}{2\times 19}$$

Multiply 2 times 19.

$$x=\frac{-20±4\sqrt{70}i}{38}$$

Now solve the equation x=\frac{-20±4\sqrt{70}i}{38} when ± is plus. Add -20 to 4i\sqrt{70}.

$$x=\frac{-20+4\sqrt{70}i}{38}$$

Divide -20+4i\sqrt{70} by 38.

$$x=\frac{-10+2\sqrt{70}i}{19}$$

Now solve the equation x=\frac{-20±4\sqrt{70}i}{38} when ± is minus. Subtract 4i\sqrt{70} from -20.

$$x=\frac{-4\sqrt{70}i-20}{38}$$

Divide -20-4i\sqrt{70} by 38.

$$x=\frac{-2\sqrt{70}i-10}{19}$$

The equation is now solved.

$$x=\frac{-10+2\sqrt{70}i}{19} x=\frac{-2\sqrt{70}i-10}{19}$$

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 35x, the least common multiple of 7x,5.

$$5\left(x+2\right)^{2}+7x\times 2x=0$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2\right)^{2}.

$$5\left(x^{2}+4x+4\right)+7x\times 2x=0$$

Use the distributive property to multiply 5 by x^{2}+4x+4.

$$5x^{2}+20x+20+7x\times 2x=0$$

Multiply x and x to get x^{2}.

$$5x^{2}+20x+20+7x^{2}\times 2=0$$

Multiply 7 and 2 to get 14.

$$5x^{2}+20x+20+14x^{2}=0$$

Combine 5x^{2} and 14x^{2} to get 19x^{2}.

$$19x^{2}+20x+20=0$$

Subtract 20 from both sides. Anything subtracted from zero gives its negation.

$$19x^{2}+20x=-20$$

Divide both sides by 19.

$$\frac{19x^{2}+20x}{19}=-\frac{20}{19}$$

Dividing by 19 undoes the multiplication by 19.

$$x^{2}+\frac{20}{19}x=-\frac{20}{19}$$

Divide \frac{20}{19}, the coefficient of the x term, by 2 to get \frac{10}{19}. Then add the square of \frac{10}{19} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{20}{19}x+\left(\frac{10}{19}\right)^{2}=-\frac{20}{19}+\left(\frac{10}{19}\right)^{2}$$

Square \frac{10}{19} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{20}{19}x+\frac{100}{361}=-\frac{20}{19}+\frac{100}{361}$$

Add -\frac{20}{19} to \frac{100}{361} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{20}{19}x+\frac{100}{361}=-\frac{280}{361}$$

Factor x^{2}+\frac{20}{19}x+\frac{100}{361}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{10}{19}\right)^{2}=-\frac{280}{361}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{10}{19}\right)^{2}}=\sqrt{-\frac{280}{361}}$$

Simplify.

$$x+\frac{10}{19}=\frac{2\sqrt{70}i}{19} x+\frac{10}{19}=-\frac{2\sqrt{70}i}{19}$$

Subtract \frac{10}{19} from both sides of the equation.

$$x=\frac{-10+2\sqrt{70}i}{19} x=\frac{-2\sqrt{70}i-10}{19}$$