Algebra Examples

$y - 5 = {(x - 2)}^{2}$ $x + 2 y = 6$

Step 1

Subtract $2 y$ from both sides of the equation.

$x = 6 - 2 y$

$y - 5 = {(x - 2)}^{2}$

Step 2

Replace all occurrences of $x$ with $6 - 2 y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $y - 5 = {(x - 2)}^{2}$ with $6 - 2 y$ .

$y - 5 = {((6 - 2 y) - 2)}^{2}$

$x = 6 - 2 y$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify ${((6 - 2 y) - 2)}^{2}$ .

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Step 2.2.1.1

Simplify the expression.

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Step 2.2.1.1.1

Subtract $2$ from $6$ .

$y - 5 = {(- 2 y + 4)}^{2}$

$x = 6 - 2 y$

Step 2.2.1.1.2

Rewrite ${(- 2 y + 4)}^{2}$ as $(- 2 y + 4) (- 2 y + 4)$ .

$y - 5 = (- 2 y + 4) (- 2 y + 4)$

$x = 6 - 2 y$

$y - 5 = (- 2 y + 4) (- 2 y + 4)$

$x = 6 - 2 y$

Step 2.2.1.2

Expand $(- 2 y + 4) (- 2 y + 4)$ using the FOIL Method.

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Step 2.2.1.2.1

Apply the distributive property.

$y - 5 = - 2 y (- 2 y + 4) + 4 (- 2 y + 4)$

$x = 6 - 2 y$

Step 2.2.1.2.2

Apply the distributive property.

$y - 5 = - 2 y (- 2 y) - 2 y \cdot 4 + 4 (- 2 y + 4)$

$x = 6 - 2 y$

Step 2.2.1.2.3

Apply the distributive property.

$y - 5 = - 2 y (- 2 y) - 2 y \cdot 4 + 4 (- 2 y) + 4 \cdot 4$

$x = 6 - 2 y$

$y - 5 = - 2 y (- 2 y) - 2 y \cdot 4 + 4 (- 2 y) + 4 \cdot 4$

$x = 6 - 2 y$

Step 2.2.1.3

Simplify and combine like terms.

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Step 2.2.1.3.1

Simplify each term.

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Step 2.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$y - 5 = - 2 \cdot (- 2 y \cdot y) - 2 y \cdot 4 + 4 (- 2 y) + 4 \cdot 4$

$x = 6 - 2 y$

Step 2.2.1.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 2.2.1.3.1.2.1

Move $y$ .

$y - 5 = - 2 \cdot (- 2 (y \cdot y)) - 2 y \cdot 4 + 4 (- 2 y) + 4 \cdot 4$

$x = 6 - 2 y$

Step 2.2.1.3.1.2.2

Multiply $y$ by $y$ .

$y - 5 = - 2 \cdot (- 2 y^{2}) - 2 y \cdot 4 + 4 (- 2 y) + 4 \cdot 4$

$x = 6 - 2 y$

$y - 5 = - 2 \cdot (- 2 y^{2}) - 2 y \cdot 4 + 4 (- 2 y) + 4 \cdot 4$

$x = 6 - 2 y$

Step 2.2.1.3.1.3

Multiply $- 2$ by $- 2$ .

$y - 5 = 4 y^{2} - 2 y \cdot 4 + 4 (- 2 y) + 4 \cdot 4$

$x = 6 - 2 y$

Step 2.2.1.3.1.4

Multiply $4$ by $- 2$ .

$y - 5 = 4 y^{2} - 8 y + 4 (- 2 y) + 4 \cdot 4$

$x = 6 - 2 y$

Step 2.2.1.3.1.5

Multiply $- 2$ by $4$ .

$y - 5 = 4 y^{2} - 8 y - 8 y + 4 \cdot 4$

$x = 6 - 2 y$

Step 2.2.1.3.1.6

Multiply $4$ by $4$ .

$y - 5 = 4 y^{2} - 8 y - 8 y + 16$

$x = 6 - 2 y$

$y - 5 = 4 y^{2} - 8 y - 8 y + 16$

$x = 6 - 2 y$

Step 2.2.1.3.2

Subtract $8 y$ from $- 8 y$ .

$y - 5 = 4 y^{2} - 16 y + 16$

$x = 6 - 2 y$

$y - 5 = 4 y^{2} - 16 y + 16$

$x = 6 - 2 y$

$y - 5 = 4 y^{2} - 16 y + 16$

$x = 6 - 2 y$

$y - 5 = 4 y^{2} - 16 y + 16$

$x = 6 - 2 y$

$y - 5 = 4 y^{2} - 16 y + 16$

$x = 6 - 2 y$

Step 3

Solve for $y$ in $y - 5 = 4 y^{2} - 16 y + 16$ .

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Step 3.1

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$4 y^{2} - 16 y + 16 = y - 5$

$x = 6 - 2 y$

Step 3.2

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

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Step 3.2.1

Subtract $y$ from both sides of the equation.

$4 y^{2} - 16 y + 16 - y = - 5$

$x = 6 - 2 y$

Step 3.2.2

Subtract $y$ from $- 16 y$ .

$4 y^{2} - 17 y + 16 = - 5$

$x = 6 - 2 y$

$4 y^{2} - 17 y + 16 = - 5$

$x = 6 - 2 y$

Step 3.3

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

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Step 3.3.1

Add $5$ to both sides of the equation.

$4 y^{2} - 17 y + 16 + 5 = 0$

$x = 6 - 2 y$

Step 3.3.2

Add $16$ and $5$ .

$4 y^{2} - 17 y + 21 = 0$

$x = 6 - 2 y$

$4 y^{2} - 17 y + 21 = 0$

$x = 6 - 2 y$

Step 3.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

$x = 6 - 2 y$

Step 3.5

Substitute the values $a = 4$ , $b = - 17$ , and $c = 21$ into the quadratic formula and solve for $y$ .

$\frac{17 \pm \sqrt{{(- 17)}^{2} - 4 \cdot (4 \cdot 21)}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.6

Simplify.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $- 17$ to the power of $2$ .

$y = \frac{17 \pm \sqrt{289 - 4 \cdot 4 \cdot 21}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.6.1.2

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

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Step 3.6.1.2.1

Multiply $- 4$ by $4$ .

$y = \frac{17 \pm \sqrt{289 - 16 \cdot 21}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.6.1.2.2

Multiply $- 16$ by $21$ .

$y = \frac{17 \pm \sqrt{289 - 336}}{2 \cdot 4}$

$x = 6 - 2 y$

$y = \frac{17 \pm \sqrt{289 - 336}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.6.1.3

Subtract $336$ from $289$ .

$y = \frac{17 \pm \sqrt{- 47}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.6.1.4

Rewrite $- 47$ as $- 1 (47)$ .

$y = \frac{17 \pm \sqrt{- 1 \cdot 47}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.6.1.5

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$y = \frac{17 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.6.1.6

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

$y = \frac{17 \pm i \sqrt{47}}{2 \cdot 4}$

$x = 6 - 2 y$

$y = \frac{17 \pm i \sqrt{47}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.6.2

Multiply $2$ by $4$ .

$y = \frac{17 \pm i \sqrt{47}}{8}$

$x = 6 - 2 y$

$y = \frac{17 \pm i \sqrt{47}}{8}$

$x = 6 - 2 y$

Step 3.7

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

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

Raise $- 17$ to the power of $2$ .

$y = \frac{17 \pm \sqrt{289 - 4 \cdot 4 \cdot 21}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.7.1.2

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

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Step 3.7.1.2.1

Multiply $- 4$ by $4$ .

$y = \frac{17 \pm \sqrt{289 - 16 \cdot 21}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.7.1.2.2

Multiply $- 16$ by $21$ .

$y = \frac{17 \pm \sqrt{289 - 336}}{2 \cdot 4}$

$x = 6 - 2 y$

$y = \frac{17 \pm \sqrt{289 - 336}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.7.1.3

Subtract $336$ from $289$ .

$y = \frac{17 \pm \sqrt{- 47}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.7.1.4

Rewrite $- 47$ as $- 1 (47)$ .

$y = \frac{17 \pm \sqrt{- 1 \cdot 47}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.7.1.5

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$y = \frac{17 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.7.1.6

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

$y = \frac{17 \pm i \sqrt{47}}{2 \cdot 4}$

$x = 6 - 2 y$

$y = \frac{17 \pm i \sqrt{47}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.7.2

Multiply $2$ by $4$ .

$y = \frac{17 \pm i \sqrt{47}}{8}$

$x = 6 - 2 y$

Step 3.7.3

Change the $\pm$ to $+$ .

$y = \frac{17 + i \sqrt{47}}{8}$

$x = 6 - 2 y$

$y = \frac{17 + i \sqrt{47}}{8}$

$x = 6 - 2 y$

Step 3.8

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

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Step 3.8.1

Simplify the numerator.

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Step 3.8.1.1

Raise $- 17$ to the power of $2$ .

$y = \frac{17 \pm \sqrt{289 - 4 \cdot 4 \cdot 21}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.8.1.2

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

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Step 3.8.1.2.1

Multiply $- 4$ by $4$ .

$y = \frac{17 \pm \sqrt{289 - 16 \cdot 21}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.8.1.2.2

Multiply $- 16$ by $21$ .

$y = \frac{17 \pm \sqrt{289 - 336}}{2 \cdot 4}$

$x = 6 - 2 y$

$y = \frac{17 \pm \sqrt{289 - 336}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.8.1.3

Subtract $336$ from $289$ .

$y = \frac{17 \pm \sqrt{- 47}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.8.1.4

Rewrite $- 47$ as $- 1 (47)$ .

$y = \frac{17 \pm \sqrt{- 1 \cdot 47}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.8.1.5

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$y = \frac{17 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.8.1.6

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

$y = \frac{17 \pm i \sqrt{47}}{2 \cdot 4}$

$x = 6 - 2 y$

$y = \frac{17 \pm i \sqrt{47}}{2 \cdot 4}$

$x = 6 - 2 y$

Step 3.8.2

Multiply $2$ by $4$ .

$y = \frac{17 \pm i \sqrt{47}}{8}$

$x = 6 - 2 y$

Step 3.8.3

Change the $\pm$ to $-$ .

$y = \frac{17 - i \sqrt{47}}{8}$

$x = 6 - 2 y$

$y = \frac{17 - i \sqrt{47}}{8}$

$x = 6 - 2 y$

Step 3.9

The final answer is the combination of both solutions.

$y = \frac{17 + i \sqrt{47}}{8}, \frac{17 - i \sqrt{47}}{8}$

$x = 6 - 2 y$

$y = \frac{17 + i \sqrt{47}}{8}, \frac{17 - i \sqrt{47}}{8}$

$x = 6 - 2 y$

Step 4

Replace all occurrences of $y$ with $\frac{17 + i \sqrt{47}}{8}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 6 - 2 y$ with $\frac{17 + i \sqrt{47}}{8}$ .

$x = 6 - 2 \frac{17 + i \sqrt{47}}{8}$

$y = \frac{17 + i \sqrt{47}}{8}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $6 - 2 \frac{17 + i \sqrt{47}}{8}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Cancel the common factor of $2$ .

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Step 4.2.1.1.1.1

Factor $2$ out of $- 2$ .

$x = 6 + 2 (- 1) (\frac{17 + i \sqrt{47}}{8})$

$y = \frac{17 + i \sqrt{47}}{8}$

Step 4.2.1.1.1.2

Factor $2$ out of $8$ .

$x = 6 + 2 \cdot (- 1 \frac{17 + i \sqrt{47}}{2 \cdot 4})$

$y = \frac{17 + i \sqrt{47}}{8}$

Step 4.2.1.1.1.3

Cancel the common factor.

$x = 6 + 2 \cdot (- 1 \frac{17 + i \sqrt{47}}{2 \cdot 4})$

$y = \frac{17 + i \sqrt{47}}{8}$

Step 4.2.1.1.1.4

Rewrite the expression.

$x = 6 - 1 \frac{17 + i \sqrt{47}}{4}$

$y = \frac{17 + i \sqrt{47}}{8}$

$x = 6 - 1 \frac{17 + i \sqrt{47}}{4}$

$y = \frac{17 + i \sqrt{47}}{8}$

Step 4.2.1.1.2

Rewrite $- 1 \frac{17 + i \sqrt{47}}{4}$ as $- \frac{17 + i \sqrt{47}}{4}$ .

$x = 6 - \frac{17 + i \sqrt{47}}{4}$

$y = \frac{17 + i \sqrt{47}}{8}$

$x = 6 - \frac{17 + i \sqrt{47}}{4}$

$y = \frac{17 + i \sqrt{47}}{8}$

Step 4.2.1.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 6 \cdot \frac{4}{4} - \frac{17 + i \sqrt{47}}{4}$

$y = \frac{17 + i \sqrt{47}}{8}$

Step 4.2.1.3

Combine $6$ and $\frac{4}{4}$ .

$x = \frac{6 \cdot 4}{4} - \frac{17 + i \sqrt{47}}{4}$

$y = \frac{17 + i \sqrt{47}}{8}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$x = \frac{7 - i \sqrt{47}}{4}$

$y = \frac{17 + i \sqrt{47}}{8}$

$x = \frac{7 - i \sqrt{47}}{4}$

$y = \frac{17 + i \sqrt{47}}{8}$

$x = \frac{7 - i \sqrt{47}}{4}$

$y = \frac{17 + i \sqrt{47}}{8}$

$x = \frac{7 - i \sqrt{47}}{4}$

$y = \frac{17 + i \sqrt{47}}{8}$

Step 5

Replace all occurrences of $y$ with $\frac{17 - i \sqrt{47}}{8}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = 6 - 2 y$ with $\frac{17 - i \sqrt{47}}{8}$ .

$x = 6 - 2 \frac{17 - i \sqrt{47}}{8}$

$y = \frac{17 - i \sqrt{47}}{8}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $6 - 2 \frac{17 - i \sqrt{47}}{8}$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Cancel the common factor of $2$ .

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Step 5.2.1.1.1.1

Factor $2$ out of $- 2$ .

$x = 6 + 2 (- 1) (\frac{17 - i \sqrt{47}}{8})$

$y = \frac{17 - i \sqrt{47}}{8}$

Step 5.2.1.1.1.2

Factor $2$ out of $8$ .

$x = 6 + 2 \cdot (- 1 \frac{17 - i \sqrt{47}}{2 \cdot 4})$

$y = \frac{17 - i \sqrt{47}}{8}$

Step 5.2.1.1.1.3

Cancel the common factor.

$x = 6 + 2 \cdot (- 1 \frac{17 - i \sqrt{47}}{2 \cdot 4})$

$y = \frac{17 - i \sqrt{47}}{8}$

Step 5.2.1.1.1.4

Rewrite the expression.

$x = 6 - 1 \frac{17 - i \sqrt{47}}{4}$

$y = \frac{17 - i \sqrt{47}}{8}$

$x = 6 - 1 \frac{17 - i \sqrt{47}}{4}$

$y = \frac{17 - i \sqrt{47}}{8}$

Step 5.2.1.1.2

Rewrite $- 1 \frac{17 - i \sqrt{47}}{4}$ as $- \frac{17 - i \sqrt{47}}{4}$ .

$x = 6 - \frac{17 - i \sqrt{47}}{4}$

$y = \frac{17 - i \sqrt{47}}{8}$

$x = 6 - \frac{17 - i \sqrt{47}}{4}$

$y = \frac{17 - i \sqrt{47}}{8}$

Step 5.2.1.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 6 \cdot \frac{4}{4} - \frac{17 - i \sqrt{47}}{4}$

$y = \frac{17 - i \sqrt{47}}{8}$

Step 5.2.1.3

Combine $6$ and $\frac{4}{4}$ .

$x = \frac{6 \cdot 4}{4} - \frac{17 - i \sqrt{47}}{4}$

$y = \frac{17 - i \sqrt{47}}{8}$

Step 5.2.1.4

Combine the numerators over the common denominator.

$x = \frac{7 + \sqrt{47} i}{4}$

$y = \frac{17 - i \sqrt{47}}{8}$

$x = \frac{7 + \sqrt{47} i}{4}$

$y = \frac{17 - i \sqrt{47}}{8}$

$x = \frac{7 + \sqrt{47} i}{4}$

$y = \frac{17 - i \sqrt{47}}{8}$

$x = \frac{7 + \sqrt{47} i}{4}$

$y = \frac{17 - i \sqrt{47}}{8}$

Step 6

List all of the solutions.

$x = \frac{7 - i \sqrt{47}}{4}, y = \frac{17 + i \sqrt{47}}{8}$

$x = \frac{7 + \sqrt{47} i}{4}, y = \frac{17 - i \sqrt{47}}{8}$

Step 7