Algebra Examples

$\sqrt{8 (x - k)} = x - k$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{8 (x - k)}}^{2} = {(x - k)}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 (x - k)}$ as ${(8 (x - k))}^{\frac{1}{2}}$ .

${({(8 (x - k))}^{\frac{1}{2}})}^{2} = {(x - k)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(8 (x - k))}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(8 (x - k))}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(8 (x - k))}^{\frac{1}{2} \cdot 2} = {(x - k)}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(8 (x - k))}^{\frac{1}{2} \cdot 2} = {(x - k)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(8 (x - k))}^{1} = {(x - k)}^{2}$

Step 2.2.1.2

Apply the distributive property.

${(8 x + 8 (- k))}^{1} = {(x - k)}^{2}$

Step 2.2.1.3

Multiply.

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Step 2.2.1.3.1

Multiply $- 1$ by $8$ .

${(8 x - 8 k)}^{1} = {(x - k)}^{2}$

Step 2.2.1.3.2

Simplify.

$8 x - 8 k = {(x - k)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

$8 x - 8 k = (x - k) (x - k)$

Step 2.3.1.2

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

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Step 2.3.1.2.1

Apply the distributive property.

$8 x - 8 k = x (x - k) - k (x - k)$

Step 2.3.1.2.2

Apply the distributive property.

$8 x - 8 k = x \cdot x + x (- k) - k (x - k)$

Step 2.3.1.2.3

Apply the distributive property.

$8 x - 8 k = x \cdot x + x (- k) - k x - k (- k)$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $x$ by $x$ .

$8 x - 8 k = x^{2} + x (- k) - k x - k (- k)$

Step 2.3.1.3.1.2

Rewrite using the commutative property of multiplication.

$8 x - 8 k = x^{2} - x k - k x - k (- k)$

Step 2.3.1.3.1.3

Rewrite using the commutative property of multiplication.

$8 x - 8 k = x^{2} - x k - k x - 1 \cdot - 1 k \cdot k$

Step 2.3.1.3.1.4

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

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Step 2.3.1.3.1.4.1

Move $k$ .

$8 x - 8 k = x^{2} - x k - k x - 1 \cdot - 1 (k \cdot k)$

Step 2.3.1.3.1.4.2

Multiply $k$ by $k$ .

$8 x - 8 k = x^{2} - x k - k x - 1 \cdot - 1 k^{2}$

Step 2.3.1.3.1.5

Multiply $- 1$ by $- 1$ .

$8 x - 8 k = x^{2} - x k - k x + 1 k^{2}$

Step 2.3.1.3.1.6

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

$8 x - 8 k = x^{2} - x k - k x + k^{2}$

Step 2.3.1.3.2

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

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Step 2.3.1.3.2.1

Move $x$ .

$8 x - 8 k = x^{2} - k x - k x + k^{2}$

Step 2.3.1.3.2.2

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

$8 x - 8 k = x^{2} - 2 k x + k^{2}$

Step 3

Solve for $k$ .

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Step 3.1

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

$x^{2} - 2 k x + k^{2} = 8 x - 8 k$

Step 3.2

Add $8 k$ to both sides of the equation.

$x^{2} - 2 k x + k^{2} + 8 k = 8 x$

Step 3.3

Subtract $8 x$ from both sides of the equation.

$x^{2} - 2 k x + k^{2} + 8 k - 8 x = 0$

Step 3.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.5

Substitute the values $a = 1$ , $b = - 2 x + 8$ , and $c = x^{2} - 8 x$ into the quadratic formula and solve for $k$ .

$\frac{- (- 2 x + 8) \pm \sqrt{{(- 2 x + 8)}^{2} - 4 \cdot (1 \cdot (x^{2} - 8 x))}}{2 \cdot 1}$

Step 3.6

Simplify.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Apply the distributive property.

$k = \frac{- (- 2 x) - 1 \cdot 8 \pm \sqrt{{(- 2 x + 8)}^{2} - 4 \cdot 1 \cdot (x^{2} - 8 x)}}{2 \cdot 1}$

Step 3.6.1.2

Multiply $- 2$ by $- 1$ .

$k = \frac{2 x - 1 \cdot 8 \pm \sqrt{{(- 2 x + 8)}^{2} - 4 \cdot 1 \cdot (x^{2} - 8 x)}}{2 \cdot 1}$

Step 3.6.1.3

Multiply $- 1$ by $8$ .

$k = \frac{2 x - 8 \pm \sqrt{{(- 2 x + 8)}^{2} - 4 \cdot 1 \cdot (x^{2} - 8 x)}}{2 \cdot 1}$

Step 3.6.1.4

Add parentheses.

$k = \frac{2 x - 8 \pm \sqrt{{(- 2 x + 8)}^{2} - 4 \cdot (1 \cdot (x^{2} - 8 x))}}{2 \cdot 1}$

Step 3.6.1.5

Let $u = 1 \cdot (x^{2} - 8 x)$ . Substitute $u$ for all occurrences of $1 \cdot (x^{2} - 8 x)$ .

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Step 3.6.1.5.1

Rewrite ${(- 2 x + 8)}^{2}$ as $(- 2 x + 8) (- 2 x + 8)$ .

$k = \frac{2 x - 8 \pm \sqrt{(- 2 x + 8) (- 2 x + 8) - 4 \cdot u}}{2 \cdot 1}$

Step 3.6.1.5.2

Expand $(- 2 x + 8) (- 2 x + 8)$ using the FOIL Method.

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Step 3.6.1.5.2.1

Apply the distributive property.

$k = \frac{2 x - 8 \pm \sqrt{- 2 x (- 2 x + 8) + 8 (- 2 x + 8) - 4 \cdot u}}{2 \cdot 1}$

Step 3.6.1.5.2.2

Apply the distributive property.

$k = \frac{2 x - 8 \pm \sqrt{- 2 x (- 2 x) - 2 x \cdot 8 + 8 (- 2 x + 8) - 4 \cdot u}}{2 \cdot 1}$

Step 3.6.1.5.2.3

Apply the distributive property.

$k = \frac{2 x - 8 \pm \sqrt{- 2 x (- 2 x) - 2 x \cdot 8 + 8 (- 2 x) + 8 \cdot 8 - 4 \cdot u}}{2 \cdot 1}$

Step 3.6.1.5.3

Simplify and combine like terms.

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Step 3.6.1.5.3.1

Simplify each term.

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Step 3.6.1.5.3.1.1

Rewrite using the commutative property of multiplication.

$k = \frac{2 x - 8 \pm \sqrt{- 2 \cdot (- 2 x \cdot x) - 2 x \cdot 8 + 8 (- 2 x) + 8 \cdot 8 - 4 \cdot u}}{2 \cdot 1}$

Step 3.6.1.5.3.1.2

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

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Step 3.6.1.5.3.1.2.1

Move $x$ .

$k = \frac{2 x - 8 \pm \sqrt{- 2 \cdot (- 2 (x \cdot x)) - 2 x \cdot 8 + 8 (- 2 x) + 8 \cdot 8 - 4 \cdot u}}{2 \cdot 1}$

Step 3.6.1.5.3.1.2.2

Multiply $x$ by $x$ .

$k = \frac{2 x - 8 \pm \sqrt{- 2 \cdot (- 2 x^{2}) - 2 x \cdot 8 + 8 (- 2 x) + 8 \cdot 8 - 4 \cdot u}}{2 \cdot 1}$

Step 3.6.1.5.3.1.3

Multiply $- 2$ by $- 2$ .

$k = \frac{2 x - 8 \pm \sqrt{4 x^{2} - 2 x \cdot 8 + 8 (- 2 x) + 8 \cdot 8 - 4 \cdot u}}{2 \cdot 1}$

Step 3.6.1.5.3.1.4

Multiply $8$ by $- 2$ .

$k = \frac{2 x - 8 \pm \sqrt{4 x^{2} - 16 x + 8 (- 2 x) + 8 \cdot 8 - 4 \cdot u}}{2 \cdot 1}$

Step 3.6.1.5.3.1.5

Multiply $- 2$ by $8$ .

$k = \frac{2 x - 8 \pm \sqrt{4 x^{2} - 16 x - 16 x + 8 \cdot 8 - 4 \cdot u}}{2 \cdot 1}$

Step 3.6.1.5.3.1.6

Multiply $8$ by $8$ .

$k = \frac{2 x - 8 \pm \sqrt{4 x^{2} - 16 x - 16 x + 64 - 4 \cdot u}}{2 \cdot 1}$

Step 3.6.1.5.3.2

Subtract $16 x$ from $- 16 x$ .

$k = \frac{2 x - 8 \pm \sqrt{4 x^{2} - 32 x + 64 - 4 \cdot u}}{2 \cdot 1}$

$k = \frac{2 x - 8 \pm \sqrt{4 x^{2} - 32 x + 64 - 4 u}}{2 \cdot 1}$

Step 3.6.1.6

Factor $4$ out of $4 x^{2} - 32 x + 64 - 4 u$ .

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Step 3.6.1.6.1

Factor $4$ out of $4 x^{2}$ .

$k = \frac{2 x - 8 \pm \sqrt{4 (x^{2}) - 32 x + 64 - 4 u}}{2 \cdot 1}$

Step 3.6.1.6.2

Factor $4$ out of $- 32 x$ .

$k = \frac{2 x - 8 \pm \sqrt{4 (x^{2}) + 4 (- 8 x) + 64 - 4 u}}{2 \cdot 1}$

Step 3.6.1.6.3

Factor $4$ out of $64$ .

$k = \frac{2 x - 8 \pm \sqrt{4 (x^{2}) + 4 (- 8 x) + 4 (16) - 4 u}}{2 \cdot 1}$

Step 3.6.1.6.4

Factor $4$ out of $- 4 u$ .

$k = \frac{2 x - 8 \pm \sqrt{4 (x^{2}) + 4 (- 8 x) + 4 (16) + 4 (- u)}}{2 \cdot 1}$

Step 3.6.1.6.5

Factor $4$ out of $4 (x^{2}) + 4 (- 8 x)$ .

$k = \frac{2 x - 8 \pm \sqrt{4 (x^{2} - 8 x) + 4 (16) + 4 (- u)}}{2 \cdot 1}$

Step 3.6.1.6.6

Factor $4$ out of $4 (x^{2} - 8 x) + 4 (16)$ .

$k = \frac{2 x - 8 \pm \sqrt{4 (x^{2} - 8 x + 16) + 4 (- u)}}{2 \cdot 1}$

Step 3.6.1.6.7

Factor $4$ out of $4 (x^{2} - 8 x + 16) + 4 (- u)$ .

$k = \frac{2 x - 8 \pm \sqrt{4 (x^{2} - 8 x + 16 - u)}}{2 \cdot 1}$

Step 3.6.1.7

Replace all occurrences of $u$ with $1 \cdot (x^{2} - 8 x)$ .

$k = \frac{2 x - 8 \pm \sqrt{4 (x^{2} - 8 x + 16 - (1 \cdot (x^{2} - 8 x)))}}{2 \cdot 1}$

Step 3.6.1.8

Simplify.

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Step 3.6.1.8.1

Simplify each term.

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Step 3.6.1.8.1.1

Multiply $x^{2} - 8 x$ by $1$ .

$k = \frac{2 x - 8 \pm \sqrt{4 (x^{2} - 8 x + 16 - (x^{2} - 8 x))}}{2 \cdot 1}$

Step 3.6.1.8.1.2

Apply the distributive property.

$k = \frac{2 x - 8 \pm \sqrt{4 (x^{2} - 8 x + 16 - x^{2} - (- 8 x))}}{2 \cdot 1}$

Step 3.6.1.8.1.3

Multiply $- 8$ by $- 1$ .

$k = \frac{2 x - 8 \pm \sqrt{4 (x^{2} - 8 x + 16 - x^{2} + 8 x)}}{2 \cdot 1}$

Step 3.6.1.8.2

Combine the opposite terms in $x^{2} - 8 x + 16 - x^{2} + 8 x$ .

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Step 3.6.1.8.2.1

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

$k = \frac{2 x - 8 \pm \sqrt{4 (- 8 x + 16 + 0 + 8 x)}}{2 \cdot 1}$

Step 3.6.1.8.2.2

Add $- 8 x + 16$ and $0$ .

$k = \frac{2 x - 8 \pm \sqrt{4 (- 8 x + 16 + 8 x)}}{2 \cdot 1}$

Step 3.6.1.8.2.3

Add $- 8 x$ and $8 x$ .

$k = \frac{2 x - 8 \pm \sqrt{4 (0 + 16)}}{2 \cdot 1}$

Step 3.6.1.8.2.4

Add $0$ and $16$ .

$k = \frac{2 x - 8 \pm \sqrt{4 \cdot 16}}{2 \cdot 1}$

Step 3.6.1.9

Multiply $4$ by $16$ .

$k = \frac{2 x - 8 \pm \sqrt{64}}{2 \cdot 1}$

Step 3.6.1.10

Rewrite $64$ as $8^{2}$ .

$k = \frac{2 x - 8 \pm \sqrt{8^{2}}}{2 \cdot 1}$

Step 3.6.1.11

Pull terms out from under the radical, assuming positive real numbers.

$k = \frac{2 x - 8 \pm 8}{2 \cdot 1}$

Step 3.6.2

Multiply $2$ by $1$ .

$k = \frac{2 x - 8 \pm 8}{2}$

Step 3.7

The final answer is the combination of both solutions.

$k = x$

$k = x - 8$

$k = x$

$k = x - 8$