Solve for x x/2+2x^2=-x(x-1)

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

Step 1

Simplify $- x (x - 1)$ .

Apply the distributive property.

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

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{x}{2} + 2 x^{2} = - (x \cdot x) - x \cdot - 1$

Multiply $x$ by $x$ .

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

Multiply $- x \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Step 2

Move all terms containing $x$ to the left side of the equation.

Add $x^{2}$ to both sides of the equation.

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

Subtract $x$ from both sides of the equation.

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

To write $- x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 x^{2} + x^{2} + \frac{x}{2} - x \cdot \frac{2}{2} = 0$

Combine $- x$ and $\frac{2}{2}$ .

$2 x^{2} + x^{2} + \frac{x}{2} + \frac{- x \cdot 2}{2} = 0$

Combine the numerators over the common denominator.

$2 x^{2} + x^{2} + \frac{x - x \cdot 2}{2} = 0$

Find the common denominator.

Write $2 x^{2}$ as a fraction with denominator $1$ .

$\frac{2 x^{2}}{1} + x^{2} + \frac{x - x \cdot 2}{2} = 0$

Multiply $\frac{2 x^{2}}{1}$ by $\frac{2}{2}$ .

$\frac{2 x^{2}}{1} \cdot \frac{2}{2} + x^{2} + \frac{x - x \cdot 2}{2} = 0$

Multiply $\frac{2 x^{2}}{1}$ by $\frac{2}{2}$ .

$\frac{2 x^{2} \cdot 2}{2} + x^{2} + \frac{x - x \cdot 2}{2} = 0$

Write $x^{2}$ as a fraction with denominator $1$ .

$\frac{2 x^{2} \cdot 2}{2} + \frac{x^{2}}{1} + \frac{x - x \cdot 2}{2} = 0$

Multiply $\frac{x^{2}}{1}$ by $\frac{2}{2}$ .

$\frac{2 x^{2} \cdot 2}{2} + \frac{x^{2}}{1} \cdot \frac{2}{2} + \frac{x - x \cdot 2}{2} = 0$

Multiply $\frac{x^{2}}{1}$ by $\frac{2}{2}$ .

$\frac{2 x^{2} \cdot 2}{2} + \frac{x^{2} \cdot 2}{2} + \frac{x - x \cdot 2}{2} = 0$

Combine the numerators over the common denominator.

$\frac{2 x^{2} \cdot 2 + x^{2} \cdot 2 + x - x \cdot 2}{2} = 0$

Simplify each term.

Multiply $2$ by $2$ .

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

Move $2$ to the left of $x^{2}$ .

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

Multiply $2$ by $- 1$ .

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

Add $4 x^{2}$ and $2 x^{2}$ .

$\frac{6 x^{2} + x - 2 x}{2} = 0$

Subtract $2 x$ from $x$ .

$\frac{6 x^{2} - x}{2} = 0$

Factor $x$ out of $6 x^{2} - x$ .

Factor $x$ out of $6 x^{2}$ .

$\frac{x (6 x) - x}{2} = 0$

Factor $x$ out of $- x$ .

$\frac{x (6 x) + x \cdot - 1}{2} = 0$

Factor $x$ out of $x (6 x) + x \cdot - 1$ .

$\frac{x (6 x - 1)}{2} = 0$

Step 3

Set the numerator equal to zero.

$x (6 x - 1) = 0$

Step 4

Solve the equation for $x$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$6 x - 1 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $6 x - 1$ equal to $0$ and solve for $x$ .

Set $6 x - 1$ equal to $0$ .

$6 x - 1 = 0$

Solve $6 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$6 x = 1$

Divide each term in $6 x = 1$ by $6$ and simplify.

Divide each term in $6 x = 1$ by $6$ .

$\frac{6 x}{6} = \frac{1}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{1}{6}$

Divide $x$ by $1$ .

$x = \frac{1}{6}$

The final solution is all the values that make $x (6 x - 1) = 0$ true.

$x = 0, \frac{1}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{1}{6}$

Decimal Form:

$x = 0, 0.1\overline{6}$

Solve for x x/2+2x^2=-x(x-1)

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