Finite Math Examples

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

Step 1

Rewrite the equation as $7 y^{2} - 4 x + 6 = 0$ .

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

Step 2

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

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Step 2.1

Add $4 x$ to both sides of the equation.

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

Step 2.2

Subtract $6$ from both sides of the equation.

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

Step 3

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

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Step 3.1

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 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $7$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

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

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

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

Step 4

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 5

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

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Step 5.1

Combine the numerators over the common denominator.

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

Step 5.2

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

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Step 5.2.1

Factor $2$ out of $4 x$ .

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

Step 5.2.2

Factor $2$ out of $- 6$ .

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

Step 5.2.3

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

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

Step 5.3

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 5.4

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 5.5

Combine and simplify the denominator.

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Step 5.5.1

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 5.5.2

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

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

Step 5.5.3

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 5.5.4

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 5.5.5

Add $1$ and $1$ .

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

Step 5.5.6

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

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Step 5.5.6.1

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 5.5.6.2

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 5.5.6.3

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

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

Step 5.5.6.4

Cancel the common factor of $2$ .

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Step 5.5.6.4.1

Cancel the common factor.

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

Step 5.5.6.4.2

Rewrite the expression.

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

Step 5.5.6.5

Evaluate the exponent.

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

Step 5.6

Simplify the numerator.

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Step 5.6.1

Combine using the product rule for radicals.

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

Step 5.6.2

Multiply $7$ by $2$ .

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

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.

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

Step 6.2

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

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

Step 6.3

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

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

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

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

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