Solve the System of Equations y=x^2+x+5 y=x+1

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

Eliminate the equal sides of each equation and combine.

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

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

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

Subtract $x$ from both sides of the equation.

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

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

Subtract $x$ from $x$ .

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

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

$x^{2} + 5 = 1$

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

Subtract $5$ from both sides of the equation.

$x^{2} = 1 - 5$

Subtract $5$ from $1$ .

$x^{2} = - 4$

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

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

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

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

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

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

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

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

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

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

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

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

$x = \pm i \cdot 2$

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

$x = \pm 2 i$

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

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

$x = 2 i$

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

$x = - 2 i$

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

$x = 2 i, - 2 i$

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

Substitute $2 i$ for $x$ .

$y = (2 i) + 1$

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

Remove parentheses.

$y = 2 i + 1$

Remove parentheses.

$y = (2 i) + 1$

Reorder $2 i$ and $1$ .

$y = 1 + 2 i$

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

Substitute $- 2 i$ for $x$ .

$y = (- 2 i) + 1$

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

Remove parentheses.

$y = - 2 i + 1$

Remove parentheses.

$y = (- 2 i) + 1$

Reorder $- 2 i$ and $1$ .

$y = 1 - 2 i$

List all of the solutions.

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

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

Solve the System of Equations y=x^2+x+5 y=x+1

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