Solve x+1/2x+2x(x+1)=0

$$x+ \frac{ 1 }{ 2 } x+2x(x+1)=0$$

Solve for x

$$x = -\frac{7}{4} = -1\frac{3}{4} = -1.75$$

$$x=0$$

Steps Using Factoring

Combine x and \frac{1}{2}x to get \frac{3}{2}x.

$$\frac{3}{2}x+2x\left(x+1\right)=0$$

Use the distributive property to multiply 2x by x+1.

$$\frac{3}{2}x+2x^{2}+2x=0$$

Combine \frac{3}{2}x and 2x to get \frac{7}{2}x.

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

Factor out x.

$$x\left(\frac{7}{2}+2x\right)=0$$

To find equation solutions, solve x=0 and \frac{7}{2}+2x=0.

$$x=0 x=-\frac{7}{4}$$

Steps Using the Quadratic Formula

Combine x and \frac{1}{2}x to get \frac{3}{2}x.

$$\frac{3}{2}x+2x\left(x+1\right)=0$$

Use the distributive property to multiply 2x by x+1.

$$\frac{3}{2}x+2x^{2}+2x=0$$

Combine \frac{3}{2}x and 2x to get \frac{7}{2}x.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, \frac{7}{2} for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{7}{2}±\sqrt{\left(\frac{7}{2}\right)^{2}}}{2\times 2}$$

Take the square root of \left(\frac{7}{2}\right)^{2}.

$$x=\frac{-\frac{7}{2}±\frac{7}{2}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-\frac{7}{2}±\frac{7}{2}}{4}$$

Now solve the equation x=\frac{-\frac{7}{2}±\frac{7}{2}}{4} when ± is plus. Add -\frac{7}{2} to \frac{7}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{0}{4}$$

Divide 0 by 4.

$$x=0$$

Now solve the equation x=\frac{-\frac{7}{2}±\frac{7}{2}}{4} when ± is minus. Subtract \frac{7}{2} from -\frac{7}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

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

The equation is now solved.

$$x=0 x=-\frac{7}{4}$$

Steps for Completing the Square

Combine x and \frac{1}{2}x to get \frac{3}{2}x.

$$\frac{3}{2}x+2x\left(x+1\right)=0$$

Use the distributive property to multiply 2x by x+1.

$$\frac{3}{2}x+2x^{2}+2x=0$$

Combine \frac{3}{2}x and 2x to get \frac{7}{2}x.

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

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.

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

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

Divide \frac{7}{2} by 2.

$$x^{2}+\frac{7}{4}x=\frac{0}{2}$$

Divide 0 by 2.

$$x^{2}+\frac{7}{4}x=0$$

Divide \frac{7}{4}, the coefficient of the x term, by 2 to get \frac{7}{8}. Then add the square of \frac{7}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{4}x+\left(\frac{7}{8}\right)^{2}=\left(\frac{7}{8}\right)^{2}$$

Square \frac{7}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{4}x+\frac{49}{64}=\frac{49}{64}$$

Factor x^{2}+\frac{7}{4}x+\frac{49}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{8}\right)^{2}=\frac{49}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{8}\right)^{2}}=\sqrt{\frac{49}{64}}$$

Simplify.

$$x+\frac{7}{8}=\frac{7}{8} x+\frac{7}{8}=-\frac{7}{8}$$

Subtract \frac{7}{8} from both sides of the equation.

$$x=0 x=-\frac{7}{4}$$