Calculus Examples

$x^{2} = 4 x - x^{2}$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$4 x - x^{2} = x^{2}$

Step 2

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

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Step 2.1

Subtract $x^{2}$ from both sides of the equation.

$4 x - x^{2} - x^{2} = 0$

Step 2.2

Subtract $x^{2}$ from $- x^{2}$ .

$4 x - 2 x^{2} = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$4 u - 2 u^{2} = 0$

Step 3.2

Factor $2 u$ out of $4 u - 2 u^{2}$ .

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Step 3.2.1

Factor $2 u$ out of $4 u$ .

$2 u (2) - 2 u^{2} = 0$

Step 3.2.2

Factor $2 u$ out of $- 2 u^{2}$ .

$2 u (2) + 2 u (- u) = 0$

Step 3.2.3

Factor $2 u$ out of $2 u (2) + 2 u (- u)$ .

$2 u (2 - u) = 0$

Step 3.3

Replace all occurrences of $u$ with $x$ .

$2 x (2 - x) = 0$

Step 4

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

$x = 0$

$2 - x = 0$

Step 5

Set $x$ equal to $0$ .

$x = 0$

Step 6

Set $2 - x$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $2 - x$ equal to $0$ .

$2 - x = 0$

Step 6.2

Solve $2 - x = 0$ for $x$ .

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Step 6.2.1

Subtract $2$ from both sides of the equation.

$- x = - 2$

Step 6.2.2

Divide each term in $- x = - 2$ by $- 1$ and simplify.

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Step 6.2.2.1

Divide each term in $- x = - 2$ by $- 1$ .

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

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 6.2.2.2.2

Divide $x$ by $1$ .

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

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Divide $- 2$ by $- 1$ .

$x = 2$

Step 7

The final solution is all the values that make $2 x (2 - x) = 0$ true.

$x = 0, 2$