Finite Math Examples

$4 x - 7 y^{2} + 6 = 0$

Step 1

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

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Step 1.1

Subtract $4 x$ from both sides of the equation.

$- 7 y^{2} + 6 = - 4 x$

Step 1.2

Subtract $6$ from both sides of the equation.

$- 7 y^{2} = - 4 x - 6$

Step 2

Divide each term in $- 7 y^{2} = - 4 x - 6$ by $- 7$ and simplify.

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Step 2.1

Divide each term in $- 7 y^{2} = - 4 x - 6$ by $- 7$ .

$\frac{- 7 y^{2}}{- 7} = \frac{- 4 x}{- 7} + \frac{- 6}{- 7}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 7$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 7 y^{2}}{- 7} = \frac{- 4 x}{- 7} + \frac{- 6}{- 7}$

Step 2.2.1.2

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

$y^{2} = \frac{- 4 x}{- 7} + \frac{- 6}{- 7}$

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

Dividing two negative values results in a positive value.

$y^{2} = \frac{4 x}{7} + \frac{- 6}{- 7}$

Step 2.3.1.2

Dividing two negative values results in a positive value.

$y^{2} = \frac{4 x}{7} + \frac{6}{7}$

Step 3

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

$y = \pm \sqrt{\frac{4 x}{7} + \frac{6}{7}}$

Step 4

Simplify $\pm \sqrt{\frac{4 x}{7} + \frac{6}{7}}$ .

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Step 4.1

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{4 x + 6}{7}}$

Step 4.2

Factor $2$ out of $4 x + 6$ .

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Step 4.2.1

Factor $2$ out of $4 x$ .

$y = \pm \sqrt{\frac{2 (2 x) + 6}{7}}$

Step 4.2.2

Factor $2$ out of $6$ .

$y = \pm \sqrt{\frac{2 (2 x) + 2 (3)}{7}}$

Step 4.2.3

Factor $2$ out of $2 (2 x) + 2 (3)$ .

$y = \pm \sqrt{\frac{2 (2 x + 3)}{7}}$

Step 4.3

Rewrite $\sqrt{\frac{2 (2 x + 3)}{7}}$ as $\frac{\sqrt{2 (2 x + 3)}}{\sqrt{7}}$ .

$y = \pm \frac{\sqrt{2 (2 x + 3)}}{\sqrt{7}}$

Step 4.4

Multiply $\frac{\sqrt{2 (2 x + 3)}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$y = \pm \frac{\sqrt{2 (2 x + 3)}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 4.5

Combine and simplify the denominator.

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Step 4.5.1

Multiply $\frac{\sqrt{2 (2 x + 3)}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$y = \pm \frac{\sqrt{2 (2 x + 3)} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 4.5.2

Raise $\sqrt{7}$ to the power of $1$ .

$y = \pm \frac{\sqrt{2 (2 x + 3)} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 4.5.3

Raise $\sqrt{7}$ to the power of $1$ .

$y = \pm \frac{\sqrt{2 (2 x + 3)} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 4.5.4

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

$y = \pm \frac{\sqrt{2 (2 x + 3)} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 4.5.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{2 (2 x + 3)} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 4.5.6

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

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Step 4.5.6.1

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

$y = \pm \frac{\sqrt{2 (2 x + 3)} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 4.5.6.2

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

$y = \pm \frac{\sqrt{2 (2 x + 3)} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 4.5.6.3

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

$y = \pm \frac{\sqrt{2 (2 x + 3)} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 4.5.6.4

Cancel the common factor of $2$ .

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Step 4.5.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{2 (2 x + 3)} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 4.5.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{2 (2 x + 3)} \sqrt{7}}{7^{1}}$

Step 4.5.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{2 (2 x + 3)} \sqrt{7}}{7}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{2 (2 x + 3) \cdot 7}}{7}$

Step 4.6.2

Multiply $7$ by $2$ .

$y = \pm \frac{\sqrt{14 (2 x + 3)}}{7}$

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.

$y = \frac{\sqrt{14 (2 x + 3)}}{7}$

Step 5.2

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

$y = - \frac{\sqrt{14 (2 x + 3)}}{7}$

Step 5.3

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

$y = \frac{\sqrt{14 (2 x + 3)}}{7}$

$y = - \frac{\sqrt{14 (2 x + 3)}}{7}$

$y = \frac{\sqrt{14 (2 x + 3)}}{7}$

$y = - \frac{\sqrt{14 (2 x + 3)}}{7}$

Step 6

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

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

Step 7

Solve for $x$ .

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Step 7.1

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

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Step 7.1.1

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

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

Step 7.1.2

Simplify the left side.

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Step 7.1.2.1

Cancel the common factor of $14$ .

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Step 7.1.2.1.1

Cancel the common factor.

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

Step 7.1.2.1.2

Divide $2 x + 3$ by $1$ .

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

Step 7.1.3

Simplify the right side.

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Step 7.1.3.1

Divide $0$ by $14$ .

$2 x + 3 \geq 0$

Step 7.2

Subtract $3$ from both sides of the inequality.

$2 x \geq - 3$

Step 7.3

Divide each term in $2 x \geq - 3$ by $2$ and simplify.

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Step 7.3.1

Divide each term in $2 x \geq - 3$ by $2$ .

$\frac{2 x}{2} \geq \frac{- 3}{2}$

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Cancel the common factor of $2$ .

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Step 7.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{- 3}{2}$

Step 7.3.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 3}{2}$

Step 7.3.3

Simplify the right side.

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Step 7.3.3.1

Move the negative in front of the fraction.

$x \geq - \frac{3}{2}$

Step 8

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

Interval Notation:

$[- \frac{3}{2}, \infty)$

Set-Builder Notation:

${x | x \geq - \frac{3}{2}}$

Step 9