Algebra Examples

$y = x^{2} - 5 x + 8$ $y = - 2 x^{2} - 13$

Step 1

Eliminate the equal sides of each equation and combine.

$x^{2} - 5 x + 8 = - 2 x^{2} - 13$

Step 2

Solve $x^{2} - 5 x + 8 = - 2 x^{2} - 13$ for $x$ .

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Step 2.1

Move all terms containing $x$ to the left side of the equation.

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Step 2.1.1

Add $2 x^{2}$ to both sides of the equation.

$x^{2} - 5 x + 8 + 2 x^{2} = - 13$

Step 2.1.2

Add $x^{2}$ and $2 x^{2}$ .

$3 x^{2} - 5 x + 8 = - 13$

Step 2.2

Move all terms to the left side of the equation and simplify.

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Step 2.2.1

Add $13$ to both sides of the equation.

$3 x^{2} - 5 x + 8 + 13 = 0$

Step 2.2.2

Add $8$ and $13$ .

$3 x^{2} - 5 x + 21 = 0$

Step 2.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} + y = - 2 x^{2} - 13$

Step 2.4

Substitute the values $a = 3$ , $b = - 5$ , and $c = 21$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (3 \cdot 21)}}{2 \cdot 3} + y = - 2 x^{2} - 13$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 3 \cdot 21}}{2 \cdot 3}$

Step 2.5.1.2

Multiply $- 4 \cdot 3 \cdot 21$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{5 \pm \sqrt{25 - 12 \cdot 21}}{2 \cdot 3}$

Step 2.5.1.2.2

Multiply $- 12$ by $21$ .

$x = \frac{5 \pm \sqrt{25 - 252}}{2 \cdot 3}$

Step 2.5.1.3

Subtract $252$ from $25$ .

$x = \frac{5 \pm \sqrt{- 227}}{2 \cdot 3}$

Step 2.5.1.4

Rewrite $- 227$ as $- 1 (227)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 227}}{2 \cdot 3}$

Step 2.5.1.5

Rewrite $\sqrt{- 1 (227)}$ as $\sqrt{- 1} \cdot \sqrt{227}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{227}}{2 \cdot 3}$

Step 2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{5 \pm i \sqrt{227}}{2 \cdot 3}$

Step 2.5.2

Multiply $2$ by $3$ .

$x = \frac{5 \pm i \sqrt{227}}{6}$

Step 2.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 3 \cdot 21}}{2 \cdot 3}$

Step 2.6.1.2

Multiply $- 4 \cdot 3 \cdot 21$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{5 \pm \sqrt{25 - 12 \cdot 21}}{2 \cdot 3}$

Step 2.6.1.2.2

Multiply $- 12$ by $21$ .

$x = \frac{5 \pm \sqrt{25 - 252}}{2 \cdot 3}$

Step 2.6.1.3

Subtract $252$ from $25$ .

$x = \frac{5 \pm \sqrt{- 227}}{2 \cdot 3}$

Step 2.6.1.4

Rewrite $- 227$ as $- 1 (227)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 227}}{2 \cdot 3}$

Step 2.6.1.5

Rewrite $\sqrt{- 1 (227)}$ as $\sqrt{- 1} \cdot \sqrt{227}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{227}}{2 \cdot 3}$

Step 2.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{5 \pm i \sqrt{227}}{2 \cdot 3}$

Step 2.6.2

Multiply $2$ by $3$ .

$x = \frac{5 \pm i \sqrt{227}}{6}$

Step 2.6.3

Change the $\pm$ to $+$ .

$x = \frac{5 + i \sqrt{227}}{6}$

Step 2.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 3 \cdot 21}}{2 \cdot 3}$

Step 2.7.1.2

Multiply $- 4 \cdot 3 \cdot 21$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{5 \pm \sqrt{25 - 12 \cdot 21}}{2 \cdot 3}$

Step 2.7.1.2.2

Multiply $- 12$ by $21$ .

$x = \frac{5 \pm \sqrt{25 - 252}}{2 \cdot 3}$

Step 2.7.1.3

Subtract $252$ from $25$ .

$x = \frac{5 \pm \sqrt{- 227}}{2 \cdot 3}$

Step 2.7.1.4

Rewrite $- 227$ as $- 1 (227)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 227}}{2 \cdot 3}$

Step 2.7.1.5

Rewrite $\sqrt{- 1 (227)}$ as $\sqrt{- 1} \cdot \sqrt{227}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{227}}{2 \cdot 3}$

Step 2.7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{5 \pm i \sqrt{227}}{2 \cdot 3}$

Step 2.7.2

Multiply $2$ by $3$ .

$x = \frac{5 \pm i \sqrt{227}}{6}$

Step 2.7.3

Change the $\pm$ to $-$ .

$x = \frac{5 - i \sqrt{227}}{6}$

Step 2.8

The final answer is the combination of both solutions.

$x = \frac{5 + i \sqrt{227}}{6}, \frac{5 - i \sqrt{227}}{6}$

Step 3

Evaluate $y$ when $x = \frac{5 + i \sqrt{227}}{6}$ .

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Step 3.1

Substitute $\frac{5 + i \sqrt{227}}{6}$ for $x$ .

$y = - 2 {(\frac{5 + i \sqrt{227}}{6})}^{2} - 13$

Step 3.2

Simplify $- 2 {(\frac{5 + i \sqrt{227}}{6})}^{2} - 13$ .

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Apply the product rule to $\frac{5 + i \sqrt{227}}{6}$ .

$y = - 2 \frac{{(5 + i \sqrt{227})}^{2}}{6^{2}} - 13$

Step 3.2.1.2

Raise $6$ to the power of $2$ .

$y = - 2 \frac{{(5 + i \sqrt{227})}^{2}}{36} - 13$

Step 3.2.1.3

Cancel the common factor of $2$ .

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Step 3.2.1.3.1

Factor $2$ out of $- 2$ .

$y = 2 (- 1) \frac{{(5 + i \sqrt{227})}^{2}}{36} - 13$

Step 3.2.1.3.2

Factor $2$ out of $36$ .

$y = 2 \cdot - 1 \frac{{(5 + i \sqrt{227})}^{2}}{2 \cdot 18} - 13$

Step 3.2.1.3.3

Cancel the common factor.

$y = 2 \cdot - 1 \frac{{(5 + i \sqrt{227})}^{2}}{2 \cdot 18} - 13$

Step 3.2.1.3.4

Rewrite the expression.

$y = - 1 \frac{{(5 + i \sqrt{227})}^{2}}{18} - 13$

Step 3.2.1.4

Rewrite ${(5 + i \sqrt{227})}^{2}$ as $(5 + i \sqrt{227}) (5 + i \sqrt{227})$ .

$y = - 1 \frac{(5 + i \sqrt{227}) (5 + i \sqrt{227})}{18} - 13$

Step 3.2.1.5

Expand $(5 + i \sqrt{227}) (5 + i \sqrt{227})$ using the FOIL Method.

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Step 3.2.1.5.1

Apply the distributive property.

$y = - 1 \frac{5 (5 + i \sqrt{227}) + i \sqrt{227} (5 + i \sqrt{227})}{18} - 13$

Step 3.2.1.5.2

Apply the distributive property.

$y = - 1 \frac{5 \cdot 5 + 5 (i \sqrt{227}) + i \sqrt{227} (5 + i \sqrt{227})}{18} - 13$

Step 3.2.1.5.3

Apply the distributive property.

$y = - 1 \frac{5 \cdot 5 + 5 (i \sqrt{227}) + i \sqrt{227} \cdot 5 + i \sqrt{227} (i \sqrt{227})}{18} - 13$

Step 3.2.1.6

Simplify and combine like terms.

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Step 3.2.1.6.1

Simplify each term.

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Step 3.2.1.6.1.1

Multiply $5$ by $5$ .

$y = - 1 \frac{25 + 5 (i \sqrt{227}) + i \sqrt{227} \cdot 5 + i \sqrt{227} (i \sqrt{227})}{18} - 13$

Step 3.2.1.6.1.2

Move $5$ to the left of $i \sqrt{227}$ .

$y = - 1 \frac{25 + 5 i \sqrt{227} + 5 \cdot (i \sqrt{227}) + i \sqrt{227} (i \sqrt{227})}{18} - 13$

Step 3.2.1.6.1.3

Multiply $i \sqrt{227} (i \sqrt{227})$ .

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Step 3.2.1.6.1.3.1

Raise $i$ to the power of $1$ .

$y = - 1 \frac{25 + 5 i \sqrt{227} + 5 i \sqrt{227} + i^{1} i \sqrt{227} \sqrt{227}}{18} - 13$

Step 3.2.1.6.1.3.2

Raise $i$ to the power of $1$ .

$y = - 1 \frac{25 + 5 i \sqrt{227} + 5 i \sqrt{227} + i^{1} i^{1} \sqrt{227} \sqrt{227}}{18} - 13$

Step 3.2.1.6.1.3.3

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

$y = - 1 \frac{25 + 5 i \sqrt{227} + 5 i \sqrt{227} + i^{1 + 1} \sqrt{227} \sqrt{227}}{18} - 13$

Step 3.2.1.6.1.3.4

Add $1$ and $1$ .

$y = - 1 \frac{25 + 5 i \sqrt{227} + 5 i \sqrt{227} + i^{2} \sqrt{227} \sqrt{227}}{18} - 13$

Step 3.2.1.6.1.3.5

Raise $\sqrt{227}$ to the power of $1$ .

$y = - 1 \frac{25 + 5 i \sqrt{227} + 5 i \sqrt{227} + i^{2} ({\sqrt{227}}^{1} \sqrt{227})}{18} - 13$

Step 3.2.1.6.1.3.6

Raise $\sqrt{227}$ to the power of $1$ .

$y = - 1 \frac{25 + 5 i \sqrt{227} + 5 i \sqrt{227} + i^{2} ({\sqrt{227}}^{1} {\sqrt{227}}^{1})}{18} - 13$

Step 3.2.1.6.1.3.7

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

$y = - 1 \frac{25 + 5 i \sqrt{227} + 5 i \sqrt{227} + i^{2} {\sqrt{227}}^{1 + 1}}{18} - 13$

Step 3.2.1.6.1.3.8

Add $1$ and $1$ .

$y = - 1 \frac{25 + 5 i \sqrt{227} + 5 i \sqrt{227} + i^{2} {\sqrt{227}}^{2}}{18} - 13$

Step 3.2.1.6.1.4

Rewrite $i^{2}$ as $- 1$ .

$y = - 1 \frac{25 + 5 i \sqrt{227} + 5 i \sqrt{227} - 1 {\sqrt{227}}^{2}}{18} - 13$

Step 3.2.1.6.1.5

Rewrite ${\sqrt{227}}^{2}$ as $227$ .

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Step 3.2.1.6.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{227}$ as $227^{\frac{1}{2}}$ .

$y = - 1 \frac{25 + 5 i \sqrt{227} + 5 i \sqrt{227} - 1 {(227^{\frac{1}{2}})}^{2}}{18} - 13$

Step 3.2.1.6.1.5.2

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

$y = - 1 \frac{25 + 5 i \sqrt{227} + 5 i \sqrt{227} - 1 \cdot 227^{\frac{1}{2} \cdot 2}}{18} - 13$

Step 3.2.1.6.1.5.3

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

$y = - 1 \frac{25 + 5 i \sqrt{227} + 5 i \sqrt{227} - 1 \cdot 227^{\frac{2}{2}}}{18} - 13$

Step 3.2.1.6.1.5.4

Cancel the common factor of $2$ .

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Step 3.2.1.6.1.5.4.1

Cancel the common factor.

$y = - 1 \frac{25 + 5 i \sqrt{227} + 5 i \sqrt{227} - 1 \cdot 227^{\frac{2}{2}}}{18} - 13$

Step 3.2.1.6.1.5.4.2

Rewrite the expression.

$y = - 1 \frac{25 + 5 i \sqrt{227} + 5 i \sqrt{227} - 1 \cdot 227^{1}}{18} - 13$

Step 3.2.1.6.1.5.5

Evaluate the exponent.

$y = - 1 \frac{25 + 5 i \sqrt{227} + 5 i \sqrt{227} - 1 \cdot 227}{18} - 13$

Step 3.2.1.6.1.6

Multiply $- 1$ by $227$ .

$y = - 1 \frac{25 + 5 i \sqrt{227} + 5 i \sqrt{227} - 227}{18} - 13$

Step 3.2.1.6.2

Subtract $227$ from $25$ .

$y = - 1 \frac{5 i \sqrt{227} + 5 i \sqrt{227} - 202}{18} - 13$

Step 3.2.1.6.3

Add $5 i \sqrt{227}$ and $5 i \sqrt{227}$ .

$y = - 1 \frac{10 i \sqrt{227} - 202}{18} - 13$

Step 3.2.1.7

Reorder $10 i \sqrt{227}$ and $- 202$ .

$y = - 1 \frac{- 202 + 10 i \sqrt{227}}{18} - 13$

Step 3.2.1.8

Cancel the common factor of $- 202 + 10 i \sqrt{227}$ and $18$ .

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Step 3.2.1.8.1

Factor $2$ out of $- 202$ .

$y = - 1 \frac{2 (- 101) + 10 i \sqrt{227}}{18} - 13$

Step 3.2.1.8.2

Factor $2$ out of $10 i \sqrt{227}$ .

$y = - 1 \frac{2 (- 101) + 2 (5 i \sqrt{227})}{18} - 13$

Step 3.2.1.8.3

Factor $2$ out of $2 (- 101) + 2 (5 i \sqrt{227})$ .

$y = - 1 \frac{2 (- 101 + 5 i \sqrt{227})}{18} - 13$

Step 3.2.1.8.4

Cancel the common factors.

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Step 3.2.1.8.4.1

Factor $2$ out of $18$ .

$y = - 1 \frac{2 (- 101 + 5 i \sqrt{227})}{2 \cdot 9} - 13$

Step 3.2.1.8.4.2

Cancel the common factor.

$y = - 1 \frac{2 (- 101 + 5 i \sqrt{227})}{2 \cdot 9} - 13$

Step 3.2.1.8.4.3

Rewrite the expression.

$y = - 1 \frac{- 101 + 5 i \sqrt{227}}{9} - 13$

Step 3.2.1.9

Rewrite $- 1 \frac{- 101 + 5 i \sqrt{227}}{9}$ as $- \frac{- 101 + 5 i \sqrt{227}}{9}$ .

$y = - \frac{- 101 + 5 i \sqrt{227}}{9} - 13$

Step 3.2.2

To write $- 13$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$y = - \frac{- 101 + 5 i \sqrt{227}}{9} - 13 \cdot \frac{9}{9}$

Step 3.2.3

Combine $- 13$ and $\frac{9}{9}$ .

$y = - \frac{- 101 + 5 i \sqrt{227}}{9} + \frac{- 13 \cdot 9}{9}$

Step 3.2.4

Combine the numerators over the common denominator.

$y = \frac{- 16 - 5 i \sqrt{227}}{9}$

Step 3.2.5

Rewrite $- 16$ as $- 1 (16)$ .

$y = \frac{- 1 (16) - 5 i \sqrt{227}}{9}$

Step 3.2.6

Factor $- 1$ out of $- 5 i \sqrt{227}$ .

$y = \frac{- 1 (16) - (5 i \sqrt{227})}{9}$

Step 3.2.7

Factor $- 1$ out of $- 1 (16) - (5 i \sqrt{227})$ .

$y = \frac{- 1 (16 + 5 i \sqrt{227})}{9}$

Step 3.2.8

Move the negative in front of the fraction.

$y = - \frac{16 + 5 i \sqrt{227}}{9}$

Step 4

Evaluate $y$ when $x = \frac{5 - i \sqrt{227}}{6}$ .

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Step 4.1

Substitute $\frac{5 - i \sqrt{227}}{6}$ for $x$ .

$y = - 2 {(\frac{5 - i \sqrt{227}}{6})}^{2} - 13$

Step 4.2

Simplify $- 2 {(\frac{5 - i \sqrt{227}}{6})}^{2} - 13$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Apply the product rule to $\frac{5 - i \sqrt{227}}{6}$ .

$y = - 2 \frac{{(5 - i \sqrt{227})}^{2}}{6^{2}} - 13$

Step 4.2.1.2

Raise $6$ to the power of $2$ .

$y = - 2 \frac{{(5 - i \sqrt{227})}^{2}}{36} - 13$

Step 4.2.1.3

Cancel the common factor of $2$ .

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Step 4.2.1.3.1

Factor $2$ out of $- 2$ .

$y = 2 (- 1) \frac{{(5 - i \sqrt{227})}^{2}}{36} - 13$

Step 4.2.1.3.2

Factor $2$ out of $36$ .

$y = 2 \cdot - 1 \frac{{(5 - i \sqrt{227})}^{2}}{2 \cdot 18} - 13$

Step 4.2.1.3.3

Cancel the common factor.

$y = 2 \cdot - 1 \frac{{(5 - i \sqrt{227})}^{2}}{2 \cdot 18} - 13$

Step 4.2.1.3.4

Rewrite the expression.

$y = - 1 \frac{{(5 - i \sqrt{227})}^{2}}{18} - 13$

Step 4.2.1.4

Rewrite ${(5 - i \sqrt{227})}^{2}$ as $(5 - i \sqrt{227}) (5 - i \sqrt{227})$ .

$y = - 1 \frac{(5 - i \sqrt{227}) (5 - i \sqrt{227})}{18} - 13$

Step 4.2.1.5

Expand $(5 - i \sqrt{227}) (5 - i \sqrt{227})$ using the FOIL Method.

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Step 4.2.1.5.1

Apply the distributive property.

$y = - 1 \frac{5 (5 - i \sqrt{227}) - i \sqrt{227} (5 - i \sqrt{227})}{18} - 13$

Step 4.2.1.5.2

Apply the distributive property.

$y = - 1 \frac{5 \cdot 5 + 5 (- i \sqrt{227}) - i \sqrt{227} (5 - i \sqrt{227})}{18} - 13$

Step 4.2.1.5.3

Apply the distributive property.

$y = - 1 \frac{5 \cdot 5 + 5 (- i \sqrt{227}) - i \sqrt{227} \cdot 5 - i \sqrt{227} (- i \sqrt{227})}{18} - 13$

Step 4.2.1.6

Simplify and combine like terms.

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Step 4.2.1.6.1

Simplify each term.

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Step 4.2.1.6.1.1

Multiply $5$ by $5$ .

$y = - 1 \frac{25 + 5 (- i \sqrt{227}) - i \sqrt{227} \cdot 5 - i \sqrt{227} (- i \sqrt{227})}{18} - 13$

Step 4.2.1.6.1.2

Multiply $- 1$ by $5$ .

$y = - 1 \frac{25 - 5 (i \sqrt{227}) - i \sqrt{227} \cdot 5 - i \sqrt{227} (- i \sqrt{227})}{18} - 13$

Step 4.2.1.6.1.3

Multiply $5$ by $- 1$ .

$y = - 1 \frac{25 - 5 i \sqrt{227} - 5 i \sqrt{227} - i \sqrt{227} (- i \sqrt{227})}{18} - 13$

Step 4.2.1.6.1.4

Multiply $- i \sqrt{227} (- i \sqrt{227})$ .

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Step 4.2.1.6.1.4.1

Multiply $- 1$ by $- 1$ .

$y = - 1 \frac{25 - 5 i \sqrt{227} - 5 i \sqrt{227} + 1 i \sqrt{227} (i \sqrt{227})}{18} - 13$

Step 4.2.1.6.1.4.2

Multiply $\sqrt{227}$ by $1$ .

$y = - 1 \frac{25 - 5 i \sqrt{227} - 5 i \sqrt{227} + \sqrt{227} i (i \sqrt{227})}{18} - 13$

Step 4.2.1.6.1.4.3

Raise $\sqrt{227}$ to the power of $1$ .

$y = - 1 \frac{25 - 5 i \sqrt{227} - 5 i \sqrt{227} + {\sqrt{227}}^{1} \sqrt{227} i i}{18} - 13$

Step 4.2.1.6.1.4.4

Raise $\sqrt{227}$ to the power of $1$ .

$y = - 1 \frac{25 - 5 i \sqrt{227} - 5 i \sqrt{227} + {\sqrt{227}}^{1} {\sqrt{227}}^{1} i i}{18} - 13$

Step 4.2.1.6.1.4.5

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

$y = - 1 \frac{25 - 5 i \sqrt{227} - 5 i \sqrt{227} + {\sqrt{227}}^{1 + 1} i i}{18} - 13$

Step 4.2.1.6.1.4.6

Add $1$ and $1$ .

$y = - 1 \frac{25 - 5 i \sqrt{227} - 5 i \sqrt{227} + {\sqrt{227}}^{2} i i}{18} - 13$

Step 4.2.1.6.1.4.7

Raise $i$ to the power of $1$ .

$y = - 1 \frac{25 - 5 i \sqrt{227} - 5 i \sqrt{227} + {\sqrt{227}}^{2} (i^{1} i)}{18} - 13$

Step 4.2.1.6.1.4.8

Raise $i$ to the power of $1$ .

$y = - 1 \frac{25 - 5 i \sqrt{227} - 5 i \sqrt{227} + {\sqrt{227}}^{2} (i^{1} i^{1})}{18} - 13$

Step 4.2.1.6.1.4.9

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

$y = - 1 \frac{25 - 5 i \sqrt{227} - 5 i \sqrt{227} + {\sqrt{227}}^{2} i^{1 + 1}}{18} - 13$

Step 4.2.1.6.1.4.10

Add $1$ and $1$ .

$y = - 1 \frac{25 - 5 i \sqrt{227} - 5 i \sqrt{227} + {\sqrt{227}}^{2} i^{2}}{18} - 13$

Step 4.2.1.6.1.5

Rewrite ${\sqrt{227}}^{2}$ as $227$ .

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Step 4.2.1.6.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{227}$ as $227^{\frac{1}{2}}$ .

$y = - 1 \frac{25 - 5 i \sqrt{227} - 5 i \sqrt{227} + {(227^{\frac{1}{2}})}^{2} i^{2}}{18} - 13$

Step 4.2.1.6.1.5.2

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

$y = - 1 \frac{25 - 5 i \sqrt{227} - 5 i \sqrt{227} + 227^{\frac{1}{2} \cdot 2} i^{2}}{18} - 13$

Step 4.2.1.6.1.5.3

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

$y = - 1 \frac{25 - 5 i \sqrt{227} - 5 i \sqrt{227} + 227^{\frac{2}{2}} i^{2}}{18} - 13$

Step 4.2.1.6.1.5.4

Cancel the common factor of $2$ .

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Step 4.2.1.6.1.5.4.1

Cancel the common factor.

$y = - 1 \frac{25 - 5 i \sqrt{227} - 5 i \sqrt{227} + 227^{\frac{2}{2}} i^{2}}{18} - 13$

Step 4.2.1.6.1.5.4.2

Rewrite the expression.

$y = - 1 \frac{25 - 5 i \sqrt{227} - 5 i \sqrt{227} + 227^{1} i^{2}}{18} - 13$

Step 4.2.1.6.1.5.5

Evaluate the exponent.

$y = - 1 \frac{25 - 5 i \sqrt{227} - 5 i \sqrt{227} + 227 i^{2}}{18} - 13$

Step 4.2.1.6.1.6

Rewrite $i^{2}$ as $- 1$ .

$y = - 1 \frac{25 - 5 i \sqrt{227} - 5 i \sqrt{227} + 227 \cdot - 1}{18} - 13$

Step 4.2.1.6.1.7

Multiply $227$ by $- 1$ .

$y = - 1 \frac{25 - 5 i \sqrt{227} - 5 i \sqrt{227} - 227}{18} - 13$

Step 4.2.1.6.2

Subtract $227$ from $25$ .

$y = - 1 \frac{- 5 i \sqrt{227} - 5 i \sqrt{227} - 202}{18} - 13$

Step 4.2.1.6.3

Subtract $5 i \sqrt{227}$ from $- 5 i \sqrt{227}$ .

$y = - 1 \frac{- 10 i \sqrt{227} - 202}{18} - 13$

Step 4.2.1.7

Reorder $- 10 i \sqrt{227}$ and $- 202$ .

$y = - 1 \frac{- 202 - 10 i \sqrt{227}}{18} - 13$

Step 4.2.1.8

Cancel the common factor of $- 202 - 10 i \sqrt{227}$ and $18$ .

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Step 4.2.1.8.1

Factor $2$ out of $- 202$ .

$y = - 1 \frac{2 (- 101) - 10 i \sqrt{227}}{18} - 13$

Step 4.2.1.8.2

Factor $2$ out of $- 10 i \sqrt{227}$ .

$y = - 1 \frac{2 (- 101) + 2 (- 5 i \sqrt{227})}{18} - 13$

Step 4.2.1.8.3

Factor $2$ out of $2 (- 101) + 2 (- 5 i \sqrt{227})$ .

$y = - 1 \frac{2 (- 101 - 5 i \sqrt{227})}{18} - 13$

Step 4.2.1.8.4

Cancel the common factors.

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Step 4.2.1.8.4.1

Factor $2$ out of $18$ .

$y = - 1 \frac{2 (- 101 - 5 i \sqrt{227})}{2 \cdot 9} - 13$

Step 4.2.1.8.4.2

Cancel the common factor.

$y = - 1 \frac{2 (- 101 - 5 i \sqrt{227})}{2 \cdot 9} - 13$

Step 4.2.1.8.4.3

Rewrite the expression.

$y = - 1 \frac{- 101 - 5 i \sqrt{227}}{9} - 13$

Step 4.2.1.9

Rewrite $- 1 \frac{- 101 - 5 i \sqrt{227}}{9}$ as $- \frac{- 101 - 5 i \sqrt{227}}{9}$ .

$y = - \frac{- 101 - 5 i \sqrt{227}}{9} - 13$

Step 4.2.2

To write $- 13$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$y = - \frac{- 101 - 5 i \sqrt{227}}{9} - 13 \cdot \frac{9}{9}$

Step 4.2.3

Combine $- 13$ and $\frac{9}{9}$ .

$y = - \frac{- 101 - 5 i \sqrt{227}}{9} + \frac{- 13 \cdot 9}{9}$

Step 4.2.4

Combine the numerators over the common denominator.

$y = \frac{- 16 + 5 i \sqrt{227}}{9}$

Step 4.2.5

Rewrite $- 16$ as $- 1 (16)$ .

$y = \frac{- 1 (16) + 5 i \sqrt{227}}{9}$

Step 4.2.6

Factor $- 1$ out of $5 i \sqrt{227}$ .

$y = \frac{- 1 (16) - (- 5 i \sqrt{227})}{9}$

Step 4.2.7

Factor $- 1$ out of $- 1 (16) - (- 5 i \sqrt{227})$ .

$y = \frac{- 1 (16 - 5 i \sqrt{227})}{9}$

Step 4.2.8

Move the negative in front of the fraction.

$y = - \frac{16 - 5 i \sqrt{227}}{9}$

Step 5

List all of the solutions.

$y = - \frac{16 + 5 i \sqrt{227}}{9}, x = \frac{5 + i \sqrt{227}}{6}$

$y = - \frac{16 - 5 i \sqrt{227}}{9}, x = \frac{5 - i \sqrt{227}}{6}$

Step 6