Linear Algebra Examples

${(3 - h)}^{2} + {(- 5 - k)}^{2} = 2 {(\sqrt{7})}^{2}$

Step 1

Subtract ${(- 5 - k)}^{2}$ from both sides of the equation.

${(3 - h)}^{2} = 2 {\sqrt{7}}^{2} - {(- 5 - k)}^{2}$

Step 2

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

${(3 - h)}^{2} = 2 {(7^{\frac{1}{2}})}^{2} - {(- 5 - k)}^{2}$

Step 2.2

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

${(3 - h)}^{2} = 2 \cdot 7^{\frac{1}{2} \cdot 2} - {(- 5 - k)}^{2}$

Step 2.3

Combine $\frac{1}{2}$ and $2$ .

${(3 - h)}^{2} = 2 \cdot 7^{\frac{2}{2}} - {(- 5 - k)}^{2}$

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Cancel the common factor.

${(3 - h)}^{2} = 2 \cdot 7^{\frac{2}{2}} - {(- 5 - k)}^{2}$

Step 2.4.2

Rewrite the expression.

${(3 - h)}^{2} = 2 \cdot 7^{1} - {(- 5 - k)}^{2}$

Step 2.5

Evaluate the exponent.

${(3 - h)}^{2} = 2 \cdot 7 - {(- 5 - k)}^{2}$

Step 3

Multiply $2$ by $7$ .

${(3 - h)}^{2} = 14 - {(- 5 - k)}^{2}$

Step 4

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

$3 - h = \pm \sqrt{14 - {(- 5 - k)}^{2}}$

Step 5

Simplify $\pm \sqrt{14 - {(- 5 - k)}^{2}}$ .

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Step 5.1

Rewrite ${(- 5 - k)}^{2}$ as $(- 5 - k) (- 5 - k)$ .

$3 - h = \pm \sqrt{14 - ((- 5 - k) (- 5 - k))}$

Step 5.2

Expand $(- 5 - k) (- 5 - k)$ using the FOIL Method.

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Step 5.2.1

Apply the distributive property.

$3 - h = \pm \sqrt{14 - (- 5 (- 5 - k) - k (- 5 - k))}$

Step 5.2.2

Apply the distributive property.

$3 - h = \pm \sqrt{14 - (- 5 \cdot - 5 - 5 (- k) - k (- 5 - k))}$

Step 5.2.3

Apply the distributive property.

$3 - h = \pm \sqrt{14 - (- 5 \cdot - 5 - 5 (- k) - k \cdot - 5 - k (- k))}$

Step 5.3

Simplify and combine like terms.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply $- 5$ by $- 5$ .

$3 - h = \pm \sqrt{14 - (25 - 5 (- k) - k \cdot - 5 - k (- k))}$

Step 5.3.1.2

Multiply $- 1$ by $- 5$ .

$3 - h = \pm \sqrt{14 - (25 + 5 k - k \cdot - 5 - k (- k))}$

Step 5.3.1.3

Multiply $- 5$ by $- 1$ .

$3 - h = \pm \sqrt{14 - (25 + 5 k + 5 k - k (- k))}$

Step 5.3.1.4

Rewrite using the commutative property of multiplication.

$3 - h = \pm \sqrt{14 - (25 + 5 k + 5 k - 1 \cdot - 1 k \cdot k)}$

Step 5.3.1.5

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

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Step 5.3.1.5.1

Move $k$ .

$3 - h = \pm \sqrt{14 - (25 + 5 k + 5 k - 1 \cdot - 1 (k \cdot k))}$

Step 5.3.1.5.2

Multiply $k$ by $k$ .

$3 - h = \pm \sqrt{14 - (25 + 5 k + 5 k - 1 \cdot - 1 k^{2})}$

Step 5.3.1.6

Multiply $- 1$ by $- 1$ .

$3 - h = \pm \sqrt{14 - (25 + 5 k + 5 k + 1 k^{2})}$

Step 5.3.1.7

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

$3 - h = \pm \sqrt{14 - (25 + 5 k + 5 k + k^{2})}$

Step 5.3.2

Add $5 k$ and $5 k$ .

$3 - h = \pm \sqrt{14 - (25 + 10 k + k^{2})}$

Step 5.4

Apply the distributive property.

$3 - h = \pm \sqrt{14 - 1 \cdot 25 - (10 k) - k^{2}}$

Step 5.5

Simplify.

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Step 5.5.1

Multiply $- 1$ by $25$ .

$3 - h = \pm \sqrt{14 - 25 - (10 k) - k^{2}}$

Step 5.5.2

Multiply $10$ by $- 1$ .

$3 - h = \pm \sqrt{14 - 25 - 10 k - k^{2}}$

Step 5.6

Subtract $25$ from $14$ .

$3 - h = \pm \sqrt{- 11 - 10 k - k^{2}}$

Step 6

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

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Step 6.1

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

$3 - h = \sqrt{- 11 - 10 k - k^{2}}$

Step 6.2

Subtract $3$ from both sides of the equation.

$- h = \sqrt{- 11 - 10 k - k^{2}} - 3$

Step 6.3

Divide each term in $- h = \sqrt{- 11 - 10 k - k^{2}} - 3$ by $- 1$ and simplify.

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Step 6.3.1

Divide each term in $- h = \sqrt{- 11 - 10 k - k^{2}} - 3$ by $- 1$ .

$\frac{- h}{- 1} = \frac{\sqrt{- 11 - 10 k - k^{2}}}{- 1} + \frac{- 3}{- 1}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Dividing two negative values results in a positive value.

$\frac{h}{1} = \frac{\sqrt{- 11 - 10 k - k^{2}}}{- 1} + \frac{- 3}{- 1}$

Step 6.3.2.2

Divide $h$ by $1$ .

$h = \frac{\sqrt{- 11 - 10 k - k^{2}}}{- 1} + \frac{- 3}{- 1}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Simplify each term.

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Step 6.3.3.1.1

Move the negative one from the denominator of $\frac{\sqrt{- 11 - 10 k - k^{2}}}{- 1}$ .

$h = - 1 \cdot \sqrt{- 11 - 10 k - k^{2}} + \frac{- 3}{- 1}$

Step 6.3.3.1.2

Rewrite $- 1 \cdot \sqrt{- 11 - 10 k - k^{2}}$ as $- \sqrt{- 11 - 10 k - k^{2}}$ .

$h = - \sqrt{- 11 - 10 k - k^{2}} + \frac{- 3}{- 1}$

Step 6.3.3.1.3

Divide $- 3$ by $- 1$ .

$h = - \sqrt{- 11 - 10 k - k^{2}} + 3$

Step 6.4

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

$3 - h = - \sqrt{- 11 - 10 k - k^{2}}$

Step 6.5

Subtract $3$ from both sides of the equation.

$- h = - \sqrt{- 11 - 10 k - k^{2}} - 3$

Step 6.6

Divide each term in $- h = - \sqrt{- 11 - 10 k - k^{2}} - 3$ by $- 1$ and simplify.

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Step 6.6.1

Divide each term in $- h = - \sqrt{- 11 - 10 k - k^{2}} - 3$ by $- 1$ .

$\frac{- h}{- 1} = \frac{- \sqrt{- 11 - 10 k - k^{2}}}{- 1} + \frac{- 3}{- 1}$

Step 6.6.2

Simplify the left side.

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Step 6.6.2.1

Dividing two negative values results in a positive value.

$\frac{h}{1} = \frac{- \sqrt{- 11 - 10 k - k^{2}}}{- 1} + \frac{- 3}{- 1}$

Step 6.6.2.2

Divide $h$ by $1$ .

$h = \frac{- \sqrt{- 11 - 10 k - k^{2}}}{- 1} + \frac{- 3}{- 1}$

Step 6.6.3

Simplify the right side.

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Step 6.6.3.1

Simplify each term.

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Step 6.6.3.1.1

Dividing two negative values results in a positive value.

$h = \frac{\sqrt{- 11 - 10 k - k^{2}}}{1} + \frac{- 3}{- 1}$

Step 6.6.3.1.2

Divide $\sqrt{- 11 - 10 k - k^{2}}$ by $1$ .

$h = \sqrt{- 11 - 10 k - k^{2}} + \frac{- 3}{- 1}$

Step 6.6.3.1.3

Divide $- 3$ by $- 1$ .

$h = \sqrt{- 11 - 10 k - k^{2}} + 3$

Step 6.7

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

$h = - \sqrt{- 11 - 10 k - k^{2}} + 3$

$h = \sqrt{- 11 - 10 k - k^{2}} + 3$

$h = - \sqrt{- 11 - 10 k - k^{2}} + 3$

$h = \sqrt{- 11 - 10 k - k^{2}} + 3$

Step 7

Set the radicand in $\sqrt{- 11 - 10 k - k^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$- 11 - 10 k - k^{2} \geq 0$

Step 8

Solve for $k$ .

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Step 8.1

Convert the inequality to an equation.

$- 11 - 10 k - k^{2} = 0$

Step 8.2

Use the quadratic formula to find the solutions.

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

Step 8.3

Substitute the values $a = - 1$ , $b = - 10$ , and $c = - 11$ into the quadratic formula and solve for $k$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (- 1 \cdot - 11)}}{2 \cdot - 1}$

Step 8.4

Simplify.

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Step 8.4.1

Simplify the numerator.

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Step 8.4.1.1

Raise $- 10$ to the power of $2$ .

$k = \frac{10 \pm \sqrt{100 - 4 \cdot - 1 \cdot - 11}}{2 \cdot - 1}$

Step 8.4.1.2

Multiply $- 4 \cdot - 1 \cdot - 11$ .

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Step 8.4.1.2.1

Multiply $- 4$ by $- 1$ .

$k = \frac{10 \pm \sqrt{100 + 4 \cdot - 11}}{2 \cdot - 1}$

Step 8.4.1.2.2

Multiply $4$ by $- 11$ .

$k = \frac{10 \pm \sqrt{100 - 44}}{2 \cdot - 1}$

Step 8.4.1.3

Subtract $44$ from $100$ .

$k = \frac{10 \pm \sqrt{56}}{2 \cdot - 1}$

Step 8.4.1.4

Rewrite $56$ as $2^{2} \cdot 14$ .

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Step 8.4.1.4.1

Factor $4$ out of $56$ .

$k = \frac{10 \pm \sqrt{4 (14)}}{2 \cdot - 1}$

Step 8.4.1.4.2

Rewrite $4$ as $2^{2}$ .

$k = \frac{10 \pm \sqrt{2^{2} \cdot 14}}{2 \cdot - 1}$

Step 8.4.1.5

Pull terms out from under the radical.

$k = \frac{10 \pm 2 \sqrt{14}}{2 \cdot - 1}$

Step 8.4.2

Multiply $2$ by $- 1$ .

$k = \frac{10 \pm 2 \sqrt{14}}{- 2}$

Step 8.4.3

Simplify $\frac{10 \pm 2 \sqrt{14}}{- 2}$ .

$k = \frac{5 \pm \sqrt{14}}{- 1}$

Step 8.4.4

Move the negative one from the denominator of $\frac{5 \pm \sqrt{14}}{- 1}$ .

$k = - 1 \cdot (5 \pm \sqrt{14})$

Step 8.4.5

Rewrite $- 1 \cdot (5 \pm \sqrt{14})$ as $- (5 \pm \sqrt{14})$ .

$k = - (5 \pm \sqrt{14})$

Step 8.5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 8.5.1

Simplify the numerator.

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Step 8.5.1.1

Raise $- 10$ to the power of $2$ .

$k = \frac{10 \pm \sqrt{100 - 4 \cdot - 1 \cdot - 11}}{2 \cdot - 1}$

Step 8.5.1.2

Multiply $- 4 \cdot - 1 \cdot - 11$ .

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Step 8.5.1.2.1

Multiply $- 4$ by $- 1$ .

$k = \frac{10 \pm \sqrt{100 + 4 \cdot - 11}}{2 \cdot - 1}$

Step 8.5.1.2.2

Multiply $4$ by $- 11$ .

$k = \frac{10 \pm \sqrt{100 - 44}}{2 \cdot - 1}$

Step 8.5.1.3

Subtract $44$ from $100$ .

$k = \frac{10 \pm \sqrt{56}}{2 \cdot - 1}$

Step 8.5.1.4

Rewrite $56$ as $2^{2} \cdot 14$ .

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Step 8.5.1.4.1

Factor $4$ out of $56$ .

$k = \frac{10 \pm \sqrt{4 (14)}}{2 \cdot - 1}$

Step 8.5.1.4.2

Rewrite $4$ as $2^{2}$ .

$k = \frac{10 \pm \sqrt{2^{2} \cdot 14}}{2 \cdot - 1}$

Step 8.5.1.5

Pull terms out from under the radical.

$k = \frac{10 \pm 2 \sqrt{14}}{2 \cdot - 1}$

Step 8.5.2

Multiply $2$ by $- 1$ .

$k = \frac{10 \pm 2 \sqrt{14}}{- 2}$

Step 8.5.3

Simplify $\frac{10 \pm 2 \sqrt{14}}{- 2}$ .

$k = \frac{5 \pm \sqrt{14}}{- 1}$

Step 8.5.4

Move the negative one from the denominator of $\frac{5 \pm \sqrt{14}}{- 1}$ .

$k = - 1 \cdot (5 \pm \sqrt{14})$

Step 8.5.5

Rewrite $- 1 \cdot (5 \pm \sqrt{14})$ as $- (5 \pm \sqrt{14})$ .

$k = - (5 \pm \sqrt{14})$

Step 8.5.6

Change the $\pm$ to $+$ .

$k = - (5 + \sqrt{14})$

Step 8.5.7

Apply the distributive property.

$k = - 1 \cdot 5 - \sqrt{14}$

Step 8.5.8

Multiply $- 1$ by $5$ .

$k = - 5 - \sqrt{14}$

Step 8.6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 8.6.1

Simplify the numerator.

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Step 8.6.1.1

Raise $- 10$ to the power of $2$ .

$k = \frac{10 \pm \sqrt{100 - 4 \cdot - 1 \cdot - 11}}{2 \cdot - 1}$

Step 8.6.1.2

Multiply $- 4 \cdot - 1 \cdot - 11$ .

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Step 8.6.1.2.1

Multiply $- 4$ by $- 1$ .

$k = \frac{10 \pm \sqrt{100 + 4 \cdot - 11}}{2 \cdot - 1}$

Step 8.6.1.2.2

Multiply $4$ by $- 11$ .

$k = \frac{10 \pm \sqrt{100 - 44}}{2 \cdot - 1}$

Step 8.6.1.3

Subtract $44$ from $100$ .

$k = \frac{10 \pm \sqrt{56}}{2 \cdot - 1}$

Step 8.6.1.4

Rewrite $56$ as $2^{2} \cdot 14$ .

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Step 8.6.1.4.1

Factor $4$ out of $56$ .

$k = \frac{10 \pm \sqrt{4 (14)}}{2 \cdot - 1}$

Step 8.6.1.4.2

Rewrite $4$ as $2^{2}$ .

$k = \frac{10 \pm \sqrt{2^{2} \cdot 14}}{2 \cdot - 1}$

Step 8.6.1.5

Pull terms out from under the radical.

$k = \frac{10 \pm 2 \sqrt{14}}{2 \cdot - 1}$

Step 8.6.2

Multiply $2$ by $- 1$ .

$k = \frac{10 \pm 2 \sqrt{14}}{- 2}$

Step 8.6.3

Simplify $\frac{10 \pm 2 \sqrt{14}}{- 2}$ .

$k = \frac{5 \pm \sqrt{14}}{- 1}$

Step 8.6.4

Move the negative one from the denominator of $\frac{5 \pm \sqrt{14}}{- 1}$ .

$k = - 1 \cdot (5 \pm \sqrt{14})$

Step 8.6.5

Rewrite $- 1 \cdot (5 \pm \sqrt{14})$ as $- (5 \pm \sqrt{14})$ .

$k = - (5 \pm \sqrt{14})$

Step 8.6.6

Change the $\pm$ to $-$ .

$k = - (5 - \sqrt{14})$

Step 8.6.7

Apply the distributive property.

$k = - 1 \cdot 5 + \sqrt{14}$

Step 8.6.8

Multiply $- 1$ by $5$ .

$k = - 5 + \sqrt{14}$

Step 8.6.9

Multiply $- - \sqrt{14}$ .

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Step 8.6.9.1

Multiply $- 1$ by $- 1$ .

$k = - 5 + 1 \sqrt{14}$

Step 8.6.9.2

Multiply $\sqrt{14}$ by $1$ .

$k = - 5 + \sqrt{14}$

Step 8.7

Consolidate the solutions.

$k = - 5 - \sqrt{14}, - 5 + \sqrt{14}$

Step 8.8

Use each root to create test intervals.

$k < - 5 - \sqrt{14}$

$- 5 - \sqrt{14} < k < - 5 + \sqrt{14}$

$k > - 5 + \sqrt{14}$

Step 8.9

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 8.9.1

Test a value on the interval $k < - 5 - \sqrt{14}$ to see if it makes the inequality true.

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Step 8.9.1.1

Choose a value on the interval $k < - 5 - \sqrt{14}$ and see if this value makes the original inequality true.

$k = - 11$

Step 8.9.1.2

Replace $k$ with $- 11$ in the original inequality.

$- 11 - 10 \cdot - 11 - {(- 11)}^{2} \geq 0$

Step 8.9.1.3

The left side $- 22$ is less than the right side $0$ , which means that the given statement is false.

False

Step 8.9.2

Test a value on the interval $- 5 - \sqrt{14} < k < - 5 + \sqrt{14}$ to see if it makes the inequality true.

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Step 8.9.2.1

Choose a value on the interval $- 5 - \sqrt{14} < k < - 5 + \sqrt{14}$ and see if this value makes the original inequality true.

$k = - 5$

Step 8.9.2.2

Replace $k$ with $- 5$ in the original inequality.

$- 11 - 10 \cdot - 5 - {(- 5)}^{2} \geq 0$

Step 8.9.2.3

The left side $14$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 8.9.3

Test a value on the interval $k > - 5 + \sqrt{14}$ to see if it makes the inequality true.

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Step 8.9.3.1

Choose a value on the interval $k > - 5 + \sqrt{14}$ and see if this value makes the original inequality true.

$k = 0$

Step 8.9.3.2

Replace $k$ with $0$ in the original inequality.

$- 11 - 10 \cdot 0 - {(0)}^{2} \geq 0$

Step 8.9.3.3

The left side $- 11$ is less than the right side $0$ , which means that the given statement is false.

False

Step 8.9.4

Compare the intervals to determine which ones satisfy the original inequality.

$k < - 5 - \sqrt{14}$ False

$- 5 - \sqrt{14} < k < - 5 + \sqrt{14}$ True

$k > - 5 + \sqrt{14}$ False

$k < - 5 - \sqrt{14}$ False

$- 5 - \sqrt{14} < k < - 5 + \sqrt{14}$ True

$k > - 5 + \sqrt{14}$ False

Step 8.10

The solution consists of all of the true intervals.

$- 5 - \sqrt{14} \leq k \leq - 5 + \sqrt{14}$

Step 9

The domain is all values of $k$ that make the expression defined.

Interval Notation:

$[- 5 - \sqrt{14}, - 5 + \sqrt{14}]$

Set-Builder Notation:

${k | - 5 - \sqrt{14} \leq k \leq - 5 + \sqrt{14}}$

Step 10