Pre-Algebra Examples

${(2 y - 3)}^{2} = 63$

Step 1

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

$2 y - 3 = \pm \sqrt{63}$

Step 2

Simplify $\pm \sqrt{63}$ .

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Step 2.1

Rewrite $63$ as $3^{2} \cdot 7$ .

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Step 2.1.1

Factor $9$ out of $63$ .

$2 y - 3 = \pm \sqrt{9 (7)}$

Step 2.1.2

Rewrite $9$ as $3^{2}$ .

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

Step 2.2

Pull terms out from under the radical.

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

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.

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

Step 3.2

Add $3$ to both sides of the equation.

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Divide $y$ by $1$ .

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

Step 3.4

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

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

Step 3.5

Add $3$ to both sides of the equation.

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

Step 3.6

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

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Step 3.6.1

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

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

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $2$ .

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Step 3.6.2.1.1

Cancel the common factor.

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

Step 3.6.2.1.2

Divide $y$ by $1$ .

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

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.

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

Step 3.7

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

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$y = 5.46862696 \dots, - 2.46862696 \dots$