Calculus Examples

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

Step 1

Rewrite the equation as $K^{2} + 2 x + x^{2} = {(x - k)}^{2}$ .

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

Step 2

Simplify ${(x - k)}^{2}$ .

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Step 2.1

Rewrite.

$K^{2} + 2 x + x^{2} = 0 + 0 + {(x - k)}^{2}$

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

Apply the distributive property.

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

Step 2.3.2

Apply the distributive property.

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

Step 2.3.3

Apply the distributive property.

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

Step 2.4

Simplify and combine like terms.

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Step 2.4.1

Simplify each term.

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Step 2.4.1.1

Multiply $x$ by $x$ .

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

Step 2.4.1.2

Rewrite using the commutative property of multiplication.

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

Step 2.4.1.3

Rewrite using the commutative property of multiplication.

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

Step 2.4.1.4

Multiply $k$ by $k$ by adding the exponents.

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Step 2.4.1.4.1

Move $k$ .

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

Step 2.4.1.4.2

Multiply $k$ by $k$ .

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

Step 2.4.1.5

Multiply $- 1$ by $- 1$ .

$K^{2} + 2 x + x^{2} = x^{2} - x k - k x + 1 k^{2}$

Step 2.4.1.6

Multiply $k^{2}$ by $1$ .

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

Step 2.4.2

Subtract $k x$ from $- x k$ .

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Step 2.4.2.1

Move $x$ .

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

Step 2.4.2.2

Subtract $k x$ from $- k x$ .

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

Step 3

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

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Step 3.1

Subtract $2 x$ from both sides of the equation.

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$K = \pm \sqrt{- 2 k x + k^{2} - 2 x}$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.1

First, use the positive value of the $\pm$ to find the first solution.

$K = \sqrt{- 2 k x + k^{2} - 2 x}$

Step 5.2

Next, use the negative value of the $\pm$ to find the second solution.

$K = - \sqrt{- 2 k x + k^{2} - 2 x}$

Step 5.3

The complete solution is the result of both the positive and negative portions of the solution.

$K = \sqrt{- 2 k x + k^{2} - 2 x}$

$K = - \sqrt{- 2 k x + k^{2} - 2 x}$

$K = \sqrt{- 2 k x + k^{2} - 2 x}$

$K = - \sqrt{- 2 k x + k^{2} - 2 x}$