Algebra Examples

$y = x^{2} + x + 5$ $y = x + 1$

Step 1

Eliminate the equal sides of each equation and combine.

$x^{2} + x + 5 = x + 1$

Step 2

Solve $x^{2} + x + 5 = x + 1$ 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

Subtract $x$ from both sides of the equation.

$x^{2} + x + 5 - x = 1$

Step 2.1.2

Combine the opposite terms in $x^{2} + x + 5 - x$ .

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Step 2.1.2.1

Subtract $x$ from $x$ .

$x^{2} + 0 + 5 = 1$

Step 2.1.2.2

Add $x^{2}$ and $0$ .

$x^{2} + 5 = 1$

Step 2.2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.2.1

Subtract $5$ from both sides of the equation.

$x^{2} = 1 - 5$

Step 2.2.2

Subtract $5$ from $1$ .

$x^{2} = - 4$

Step 2.3

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

$x = \pm \sqrt{- 4}$

Step 2.4

Simplify $\pm \sqrt{- 4}$ .

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Step 2.4.1

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Step 2.4.2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{4}$

Step 2.4.3

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

$x = \pm i \cdot \sqrt{4}$

Step 2.4.4

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

$x = \pm i \cdot \sqrt{2^{2}}$

Step 2.4.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \cdot 2$

Step 2.4.6

Move $2$ to the left of $i$ .

$x = \pm 2 i$

Step 2.5

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

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Step 2.5.1

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

$x = 2 i$

Step 2.5.2

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

$x = - 2 i$

Step 2.5.3

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

$x = 2 i, - 2 i$

Step 3

Evaluate $y$ when $x = 2 i$ .

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Step 3.1

Substitute $2 i$ for $x$ .

$y = (2 i) + 1$

Step 3.2

Substitute $2 i$ for $x$ in $y = (2 i) + 1$ and solve for $y$ .

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Step 3.2.1

Remove parentheses.

$y = 2 i + 1$

Step 3.2.2

Remove parentheses.

$y = (2 i) + 1$

Step 3.2.3

Reorder $2 i$ and $1$ .

$y = 1 + 2 i$

Step 4

Evaluate $y$ when $x = - 2 i$ .

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Step 4.1

Substitute $- 2 i$ for $x$ .

$y = (- 2 i) + 1$

Step 4.2

Substitute $- 2 i$ for $x$ in $y = (- 2 i) + 1$ and solve for $y$ .

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Step 4.2.1

Remove parentheses.

$y = - 2 i + 1$

Step 4.2.2

Remove parentheses.

$y = (- 2 i) + 1$

Step 4.2.3

Reorder $- 2 i$ and $1$ .

$y = 1 - 2 i$

Step 5

List all of the solutions.

$y = 1 + 2 i, x = 2 i$

$y = 1 - 2 i, x = - 2 i$

Step 6