Basic Math Examples

$\frac{1}{2} \cdot {(2 y - 3)}^{2} + 1 = 11$

Step 1

Subtract $1$ from both sides of the equation.

$\frac{1}{2} \cdot {(2 y - 3)}^{2} = 11 - 1$

Step 2

Combine $\frac{1}{2}$ and ${(2 y - 3)}^{2}$ .

$\frac{{(2 y - 3)}^{2}}{2} = 11 - 1$

Step 3

Subtract $1$ from $11$ .

$\frac{{(2 y - 3)}^{2}}{2} = 10$

Step 4

Multiply both sides by $2$ .

$\frac{{(2 y - 3)}^{2}}{2} \cdot 2 = 10 \cdot 2$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $2$ .

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Step 5.1.1.1

Cancel the common factor.

$\frac{{(2 y - 3)}^{2}}{2} \cdot 2 = 10 \cdot 2$

Step 5.1.1.2

Rewrite the expression.

${(2 y - 3)}^{2} = 10 \cdot 2$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $10$ by $2$ .

${(2 y - 3)}^{2} = 20$

Step 6

Solve for $y$ .

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Step 6.1

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

$2 y - 3 = \pm \sqrt{20}$

Step 6.2

Simplify $\pm \sqrt{20}$ .

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Step 6.2.1

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 6.2.1.1

Factor $4$ out of $20$ .

$2 y - 3 = \pm \sqrt{4 (5)}$

Step 6.2.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 6.2.2

Pull terms out from under the radical.

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

Step 6.3

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

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Step 6.3.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 6.3.2

Add $3$ to both sides of the equation.

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

Step 6.3.3

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

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Step 6.3.3.1

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

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

Step 6.3.3.2

Simplify the left side.

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Step 6.3.3.2.1

Cancel the common factor of $2$ .

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Step 6.3.3.2.1.1

Cancel the common factor.

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

Step 6.3.3.2.1.2

Divide $y$ by $1$ .

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

Step 6.3.3.3

Simplify the right side.

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Step 6.3.3.3.1

Cancel the common factor of $2$ .

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Step 6.3.3.3.1.1

Cancel the common factor.

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

Step 6.3.3.3.1.2

Divide $\sqrt{5}$ by $1$ .

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

Step 6.3.4

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

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

Step 6.3.5

Add $3$ to both sides of the equation.

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

Step 6.3.6

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

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Step 6.3.6.1

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

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

Step 6.3.6.2

Simplify the left side.

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Step 6.3.6.2.1

Cancel the common factor of $2$ .

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Step 6.3.6.2.1.1

Cancel the common factor.

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

Step 6.3.6.2.1.2

Divide $y$ by $1$ .

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

Step 6.3.6.3

Simplify the right side.

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Step 6.3.6.3.1

Cancel the common factor of $- 2$ and $2$ .

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Step 6.3.6.3.1.1

Factor $2$ out of $- 2 \sqrt{5}$ .

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

Step 6.3.6.3.1.2

Cancel the common factors.

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Step 6.3.6.3.1.2.1

Factor $2$ out of $2$ .

$y = \frac{2 (- \sqrt{5})}{2 (1)} + \frac{3}{2}$

Step 6.3.6.3.1.2.2

Cancel the common factor.

$y = \frac{2 (- \sqrt{5})}{2 \cdot 1} + \frac{3}{2}$

Step 6.3.6.3.1.2.3

Rewrite the expression.

$y = \frac{- \sqrt{5}}{1} + \frac{3}{2}$

Step 6.3.6.3.1.2.4

Divide $- \sqrt{5}$ by $1$ .

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

Step 6.3.7

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

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$y = 3.73606797 \dots, - 0.73606797 \dots$