Finite Math Examples

${(7 y - 2)}^{2} = 75$

Step 1

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

$7 y - 2 = \pm \sqrt{75}$

Step 2

Simplify $\pm \sqrt{75}$ .

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Step 2.1

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 2.1.1

Factor $25$ out of $75$ .

$7 y - 2 = \pm \sqrt{25 (3)}$

Step 2.1.2

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

$7 y - 2 = \pm \sqrt{5^{2} \cdot 3}$

Step 2.2

Pull terms out from under the radical.

$7 y - 2 = \pm 5 \sqrt{3}$

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.

$7 y - 2 = 5 \sqrt{3}$

Step 3.2

Add $2$ to both sides of the equation.

$7 y = 5 \sqrt{3} + 2$

Step 3.3

Divide each term in $7 y = 5 \sqrt{3} + 2$ by $7$ and simplify.

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Step 3.3.1

Divide each term in $7 y = 5 \sqrt{3} + 2$ by $7$ .

$\frac{7 y}{7} = \frac{5 \sqrt{3}}{7} + \frac{2}{7}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $7$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{7 y}{7} = \frac{5 \sqrt{3}}{7} + \frac{2}{7}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{5 \sqrt{3}}{7} + \frac{2}{7}$

Step 3.4

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

$7 y - 2 = - 5 \sqrt{3}$

Step 3.5

Add $2$ to both sides of the equation.

$7 y = - 5 \sqrt{3} + 2$

Step 3.6

Divide each term in $7 y = - 5 \sqrt{3} + 2$ by $7$ and simplify.

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Step 3.6.1

Divide each term in $7 y = - 5 \sqrt{3} + 2$ by $7$ .

$\frac{7 y}{7} = \frac{- 5 \sqrt{3}}{7} + \frac{2}{7}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $7$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{7 y}{7} = \frac{- 5 \sqrt{3}}{7} + \frac{2}{7}$

Step 3.6.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 5 \sqrt{3}}{7} + \frac{2}{7}$

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

Move the negative in front of the fraction.

$y = - \frac{5 \sqrt{3}}{7} + \frac{2}{7}$

Step 3.7

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

$y = \frac{5 \sqrt{3}}{7} + \frac{2}{7}, - \frac{5 \sqrt{3}}{7} + \frac{2}{7}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$y = \frac{5 \sqrt{3}}{7} + \frac{2}{7}, - \frac{5 \sqrt{3}}{7} + \frac{2}{7}$

Decimal Form:

$y = 1.52289343 \dots, - 0.95146486 \dots$