Finite Math Examples

$2 k^{2} - 14 = - 3 x$

Step 1

Add $14$ to both sides of the equation.

$2 k^{2} = - 3 x + 14$

Step 2

Divide each term in $2 k^{2} = - 3 x + 14$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 k^{2} = - 3 x + 14$ by $2$ .

$\frac{2 k^{2}}{2} = \frac{- 3 x}{2} + \frac{14}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 k^{2}}{2} = \frac{- 3 x}{2} + \frac{14}{2}$

Step 2.2.1.2

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

$k^{2} = \frac{- 3 x}{2} + \frac{14}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Move the negative in front of the fraction.

$k^{2} = - \frac{3 x}{2} + \frac{14}{2}$

Step 2.3.1.2

Divide $14$ by $2$ .

$k^{2} = - \frac{3 x}{2} + 7$

Step 3

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

$k = \pm \sqrt{- \frac{3 x}{2} + 7}$

Step 4

Simplify $\pm \sqrt{- \frac{3 x}{2} + 7}$ .

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Step 4.1

To write $7$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$k = \pm \sqrt{- \frac{3 x}{2} + 7 \cdot \frac{2}{2}}$

Step 4.2

Combine $7$ and $\frac{2}{2}$ .

$k = \pm \sqrt{- \frac{3 x}{2} + \frac{7 \cdot 2}{2}}$

Step 4.3

Combine the numerators over the common denominator.

$k = \pm \sqrt{\frac{- 3 x + 7 \cdot 2}{2}}$

Step 4.4

Multiply $7$ by $2$ .

$k = \pm \sqrt{\frac{- 3 x + 14}{2}}$

Step 4.5

Rewrite $\sqrt{\frac{- 3 x + 14}{2}}$ as $\frac{\sqrt{- 3 x + 14}}{\sqrt{2}}$ .

$k = \pm \frac{\sqrt{- 3 x + 14}}{\sqrt{2}}$

Step 4.6

Multiply $\frac{\sqrt{- 3 x + 14}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$k = \pm \frac{\sqrt{- 3 x + 14}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 4.7

Combine and simplify the denominator.

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Step 4.7.1

Multiply $\frac{\sqrt{- 3 x + 14}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$k = \pm \frac{\sqrt{- 3 x + 14} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 4.7.2

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

$k = \pm \frac{\sqrt{- 3 x + 14} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 4.7.3

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

$k = \pm \frac{\sqrt{- 3 x + 14} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 4.7.4

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

$k = \pm \frac{\sqrt{- 3 x + 14} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 4.7.5

Add $1$ and $1$ .

$k = \pm \frac{\sqrt{- 3 x + 14} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 4.7.6

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

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Step 4.7.6.1

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

$k = \pm \frac{\sqrt{- 3 x + 14} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 4.7.6.2

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

$k = \pm \frac{\sqrt{- 3 x + 14} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 4.7.6.3

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

$k = \pm \frac{\sqrt{- 3 x + 14} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.7.6.4

Cancel the common factor of $2$ .

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Step 4.7.6.4.1

Cancel the common factor.

$k = \pm \frac{\sqrt{- 3 x + 14} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.7.6.4.2

Rewrite the expression.

$k = \pm \frac{\sqrt{- 3 x + 14} \sqrt{2}}{2^{1}}$

Step 4.7.6.5

Evaluate the exponent.

$k = \pm \frac{\sqrt{- 3 x + 14} \sqrt{2}}{2}$

Step 4.8

Combine using the product rule for radicals.

$k = \pm \frac{\sqrt{(- 3 x + 14) \cdot 2}}{2}$

Step 4.9

Reorder factors in $\pm \frac{\sqrt{(- 3 x + 14) \cdot 2}}{2}$ .

$k = \pm \frac{\sqrt{2 (- 3 x + 14)}}{2}$

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.

$k = \frac{\sqrt{2 (- 3 x + 14)}}{2}$

Step 5.2

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

$k = - \frac{\sqrt{2 (- 3 x + 14)}}{2}$

Step 5.3

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

$k = \frac{\sqrt{2 (- 3 x + 14)}}{2}$

$k = - \frac{\sqrt{2 (- 3 x + 14)}}{2}$

$k = \frac{\sqrt{2 (- 3 x + 14)}}{2}$

$k = - \frac{\sqrt{2 (- 3 x + 14)}}{2}$

Step 6

Set the radicand in $\sqrt{2 (- 3 x + 14)}$ greater than or equal to $0$ to find where the expression is defined.

$2 (- 3 x + 14) \geq 0$

Step 7

Solve for $x$ .

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Step 7.1

Divide each term in $2 (- 3 x + 14) \geq 0$ by $2$ and simplify.

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Step 7.1.1

Divide each term in $2 (- 3 x + 14) \geq 0$ by $2$ .

$\frac{2 (- 3 x + 14)}{2} \geq \frac{0}{2}$

Step 7.1.2

Simplify the left side.

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Step 7.1.2.1

Cancel the common factor of $2$ .

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Step 7.1.2.1.1

Cancel the common factor.

$\frac{2 (- 3 x + 14)}{2} \geq \frac{0}{2}$

Step 7.1.2.1.2

Divide $- 3 x + 14$ by $1$ .

$- 3 x + 14 \geq \frac{0}{2}$

Step 7.1.3

Simplify the right side.

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Step 7.1.3.1

Divide $0$ by $2$ .

$- 3 x + 14 \geq 0$

Step 7.2

Subtract $14$ from both sides of the inequality.

$- 3 x \geq - 14$

Step 7.3

Divide each term in $- 3 x \geq - 14$ by $- 3$ and simplify.

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Step 7.3.1

Divide each term in $- 3 x \geq - 14$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 x}{- 3} \leq \frac{- 14}{- 3}$

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Cancel the common factor of $- 3$ .

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Step 7.3.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} \leq \frac{- 14}{- 3}$

Step 7.3.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 14}{- 3}$

Step 7.3.3

Simplify the right side.

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Step 7.3.3.1

Dividing two negative values results in a positive value.

$x \leq \frac{14}{3}$

Step 8

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

Interval Notation:

$(- \infty, \frac{14}{3}]$

Set-Builder Notation:

${x | x \leq \frac{14}{3}}$

Step 9