Calculus Examples

${(x + k)}^{2} = x^{2} - 2 x + k^{2}$

Step 1

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

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Step 1.1

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

${(x + k)}^{2} - k^{2} = x^{2} - 2 x$

Step 1.2

Simplify each term.

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Step 1.2.1

Rewrite ${(x + k)}^{2}$ as $(x + k) (x + k)$ .

$(x + k) (x + k) - k^{2} = x^{2} - 2 x$

Step 1.2.2

Expand $(x + k) (x + k)$ using the FOIL Method.

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Step 1.2.2.1

Apply the distributive property.

$x (x + k) + k (x + k) - k^{2} = x^{2} - 2 x$

Step 1.2.2.2

Apply the distributive property.

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

Step 1.2.2.3

Apply the distributive property.

$x \cdot x + x k + k x + k \cdot k - k^{2} = x^{2} - 2 x$

Step 1.2.3

Simplify and combine like terms.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x k + k x + k \cdot k - k^{2} = x^{2} - 2 x$

Step 1.2.3.1.2

Multiply $k$ by $k$ .

$x^{2} + x k + k x + k^{2} - k^{2} = x^{2} - 2 x$

Step 1.2.3.2

Add $x k$ and $k x$ .

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Step 1.2.3.2.1

Reorder $x$ and $k$ .

$x^{2} + k x + k x + k^{2} - k^{2} = x^{2} - 2 x$

Step 1.2.3.2.2

Add $k x$ and $k x$ .

$x^{2} + 2 k x + k^{2} - k^{2} = x^{2} - 2 x$

Step 1.3

Combine the opposite terms in $x^{2} + 2 k x + k^{2} - k^{2}$ .

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Step 1.3.1

Subtract $k^{2}$ from $k^{2}$ .

$x^{2} + 2 k x + 0 = x^{2} - 2 x$

Step 1.3.2

Add $x^{2} + 2 k x$ and $0$ .

$x^{2} + 2 k x = x^{2} - 2 x$

Step 2

Move all terms not containing $k$ to the right side of the equation.

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Step 2.1

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

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

Step 2.2

Combine the opposite terms in $x^{2} - 2 x - x^{2}$ .

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Step 2.2.1

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

$2 k x = - 2 x + 0$

Step 2.2.2

Add $- 2 x$ and $0$ .

$2 k x = - 2 x$

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Rewrite the expression.

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

Step 3.2.2

Cancel the common factor of $x$ .

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Step 3.2.2.1

Cancel the common factor.

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

Step 3.2.2.2

Divide $k$ by $1$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $- 2$ and $2$ .

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Step 3.3.1.1

Factor $2$ out of $- 2 x$ .

$k = \frac{2 (- x)}{2 x}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $2$ out of $2 x$ .

$k = \frac{2 (- x)}{2 (x)}$

Step 3.3.1.2.2

Cancel the common factor.

$k = \frac{2 (- x)}{2 x}$

Step 3.3.1.2.3

Rewrite the expression.

$k = \frac{- x}{x}$

Step 3.3.2

Cancel the common factor of $x$ .

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Step 3.3.2.1

Cancel the common factor.

$k = \frac{- x}{x}$

Step 3.3.2.2

Divide $- 1$ by $1$ .

$k = - 1$