Linear Algebra Examples

$z^{4} - 50 x^{2} + 49 = 0$

Step 1

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

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Step 1.1

Add $50 x^{2}$ to both sides of the equation.

$z^{4} + 49 = 50 x^{2}$

Step 1.2

Subtract $49$ from both sides of the equation.

$z^{4} = 50 x^{2} - 49$

Step 2

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

$z = \pm \sqrt[4]{50 x^{2} - 49}$

Step 3

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

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Step 3.1

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

$z = \sqrt[4]{50 x^{2} - 49}$

Step 3.2

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

$z = - \sqrt[4]{50 x^{2} - 49}$

Step 3.3

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

$z = \sqrt[4]{50 x^{2} - 49}$

$z = - \sqrt[4]{50 x^{2} - 49}$

$z = \sqrt[4]{50 x^{2} - 49}$

$z = - \sqrt[4]{50 x^{2} - 49}$

Step 4

Set the radicand in $\sqrt[4]{50 x^{2} - 49}$ greater than or equal to $0$ to find where the expression is defined.

$50 x^{2} - 49 \geq 0$

Step 5

Solve for $x$ .

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Step 5.1

Add $49$ to both sides of the inequality.

$50 x^{2} \geq 49$

Step 5.2

Divide each term in $50 x^{2} \geq 49$ by $50$ and simplify.

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Step 5.2.1

Divide each term in $50 x^{2} \geq 49$ by $50$ .

$\frac{50 x^{2}}{50} \geq \frac{49}{50}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $50$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{50 x^{2}}{50} \geq \frac{49}{50}$

Step 5.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} \geq \frac{49}{50}$

Step 5.3

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

$\sqrt{x^{2}} \geq \sqrt{\frac{49}{50}}$

Step 5.4

Simplify the equation.

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Step 5.4.1

Simplify the left side.

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Step 5.4.1.1

Pull terms out from under the radical.

$| x | \geq \sqrt{\frac{49}{50}}$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Simplify $\sqrt{\frac{49}{50}}$ .

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Step 5.4.2.1.1

Rewrite $\sqrt{\frac{49}{50}}$ as $\frac{\sqrt{49}}{\sqrt{50}}$ .

$| x | \geq \frac{\sqrt{49}}{\sqrt{50}}$

Step 5.4.2.1.2

Simplify the numerator.

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Step 5.4.2.1.2.1

Rewrite $49$ as $7^{2}$ .

$| x | \geq \frac{\sqrt{7^{2}}}{\sqrt{50}}$

Step 5.4.2.1.2.2

Pull terms out from under the radical.

$| x | \geq \frac{| 7 |}{\sqrt{50}}$

Step 5.4.2.1.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $7$ is $7$ .

$| x | \geq \frac{7}{\sqrt{50}}$

Step 5.4.2.1.3

Simplify the denominator.

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Step 5.4.2.1.3.1

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 5.4.2.1.3.1.1

Factor $25$ out of $50$ .

$| x | \geq \frac{7}{\sqrt{25 (2)}}$

Step 5.4.2.1.3.1.2

Rewrite $25$ as $5^{2}$ .

$| x | \geq \frac{7}{\sqrt{5^{2} \cdot 2}}$

Step 5.4.2.1.3.2

Pull terms out from under the radical.

$| x | \geq \frac{7}{| 5 | \sqrt{2}}$

Step 5.4.2.1.3.3

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$| x | \geq \frac{7}{5 \sqrt{2}}$

Step 5.4.2.1.4

Multiply $\frac{7}{5 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$| x | \geq \frac{7}{5 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 5.4.2.1.5

Combine and simplify the denominator.

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Step 5.4.2.1.5.1

Multiply $\frac{7}{5 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$| x | \geq \frac{7 \sqrt{2}}{5 \sqrt{2} \sqrt{2}}$

Step 5.4.2.1.5.2

Move $\sqrt{2}$ .

$| x | \geq \frac{7 \sqrt{2}}{5 (\sqrt{2} \sqrt{2})}$

Step 5.4.2.1.5.3

Raise $\sqrt{2}$ to the power of $1$ .

$| x | \geq \frac{7 \sqrt{2}}{5 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 5.4.2.1.5.4

Raise $\sqrt{2}$ to the power of $1$ .

$| x | \geq \frac{7 \sqrt{2}}{5 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 5.4.2.1.5.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$| x | \geq \frac{7 \sqrt{2}}{5 {\sqrt{2}}^{1 + 1}}$

Step 5.4.2.1.5.6

Add $1$ and $1$ .

$| x | \geq \frac{7 \sqrt{2}}{5 {\sqrt{2}}^{2}}$

Step 5.4.2.1.5.7

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

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Step 5.4.2.1.5.7.1

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

$| x | \geq \frac{7 \sqrt{2}}{5 {(2^{\frac{1}{2}})}^{2}}$

Step 5.4.2.1.5.7.2

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

$| x | \geq \frac{7 \sqrt{2}}{5 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 5.4.2.1.5.7.3

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

$| x | \geq \frac{7 \sqrt{2}}{5 \cdot 2^{\frac{2}{2}}}$

Step 5.4.2.1.5.7.4

Cancel the common factor of $2$ .

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Step 5.4.2.1.5.7.4.1

Cancel the common factor.

$| x | \geq \frac{7 \sqrt{2}}{5 \cdot 2^{\frac{2}{2}}}$

Step 5.4.2.1.5.7.4.2

Rewrite the expression.

$| x | \geq \frac{7 \sqrt{2}}{5 \cdot 2^{1}}$

Step 5.4.2.1.5.7.5

Evaluate the exponent.

$| x | \geq \frac{7 \sqrt{2}}{5 \cdot 2}$

Step 5.4.2.1.6

Multiply $5$ by $2$ .

$| x | \geq \frac{7 \sqrt{2}}{10}$

Step 5.5

Write $| x | \geq \frac{7 \sqrt{2}}{10}$ as a piecewise.

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Step 5.5.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 5.5.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \geq \frac{7 \sqrt{2}}{10}$

Step 5.5.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 5.5.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \geq \frac{7 \sqrt{2}}{10}$

Step 5.5.5

Write as a piecewise.

${\begin{cases} x \geq \frac{7 \sqrt{2}}{10} & x \geq 0 \\ - x \geq \frac{7 \sqrt{2}}{10} & x < 0 \end{cases}$

Step 5.6

Find the intersection of $x \geq \frac{7 \sqrt{2}}{10}$ and $x \geq 0$ .

$x \geq \frac{7 \sqrt{2}}{10}$

Step 5.7

Divide each term in $- x \geq \frac{7 \sqrt{2}}{10}$ by $- 1$ and simplify.

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Step 5.7.1

Divide each term in $- x \geq \frac{7 \sqrt{2}}{10}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{\frac{7 \sqrt{2}}{10}}{- 1}$

Step 5.7.2

Simplify the left side.

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Step 5.7.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{\frac{7 \sqrt{2}}{10}}{- 1}$

Step 5.7.2.2

Divide $x$ by $1$ .

$x \leq \frac{\frac{7 \sqrt{2}}{10}}{- 1}$

Step 5.7.3

Simplify the right side.

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Step 5.7.3.1

Move the negative one from the denominator of $\frac{\frac{7 \sqrt{2}}{10}}{- 1}$ .

$x \leq - 1 \cdot \frac{7 \sqrt{2}}{10}$

Step 5.7.3.2

Rewrite $- 1 \cdot \frac{7 \sqrt{2}}{10}$ as $- \frac{7 \sqrt{2}}{10}$ .

$x \leq - \frac{7 \sqrt{2}}{10}$

Step 5.8

Find the union of the solutions.

$x \leq - \frac{7 \sqrt{2}}{10}$ or $x \geq \frac{7 \sqrt{2}}{10}$

Step 6

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

Interval Notation:

$(- \infty, - \frac{7 \sqrt{2}}{10}] \cup [\frac{7 \sqrt{2}}{10}, \infty)$

Set-Builder Notation:

${x | x \leq - \frac{7 \sqrt{2}}{10}, x \geq \frac{7 \sqrt{2}}{10}}$

Step 7