Algebra Examples

$x + 4 = x^{2}$

Step 1

Subtract $x^{2}$ from both sides of the equation.

$x + 4 - x^{2} = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = - 1$ , $b = 1$ , and $c = 4$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (- 1 \cdot 4)}}{2 \cdot - 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 1 \cdot 4}}{2 \cdot - 1}$

Step 4.1.2

Multiply $- 4 \cdot - 1 \cdot 4$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 1 \pm \sqrt{1 + 4 \cdot 4}}{2 \cdot - 1}$

Step 4.1.2.2

Multiply $4$ by $4$ .

$x = \frac{- 1 \pm \sqrt{1 + 16}}{2 \cdot - 1}$

Step 4.1.3

Add $1$ and $16$ .

$x = \frac{- 1 \pm \sqrt{17}}{2 \cdot - 1}$

Step 4.2

Multiply $2$ by $- 1$ .

$x = \frac{- 1 \pm \sqrt{17}}{- 2}$

Step 4.3

Simplify $\frac{- 1 \pm \sqrt{17}}{- 2}$ .

$x = \frac{1 \pm \sqrt{17}}{2}$

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 1 \cdot 4}}{2 \cdot - 1}$

Step 5.1.2

Multiply $- 4 \cdot - 1 \cdot 4$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 1 \pm \sqrt{1 + 4 \cdot 4}}{2 \cdot - 1}$

Step 5.1.2.2

Multiply $4$ by $4$ .

$x = \frac{- 1 \pm \sqrt{1 + 16}}{2 \cdot - 1}$

Step 5.1.3

Add $1$ and $16$ .

$x = \frac{- 1 \pm \sqrt{17}}{2 \cdot - 1}$

Step 5.2

Multiply $2$ by $- 1$ .

$x = \frac{- 1 \pm \sqrt{17}}{- 2}$

Step 5.3

Simplify $\frac{- 1 \pm \sqrt{17}}{- 2}$ .

$x = \frac{1 \pm \sqrt{17}}{2}$

Step 5.4

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{17}}{2}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 1 \cdot 4}}{2 \cdot - 1}$

Step 6.1.2

Multiply $- 4 \cdot - 1 \cdot 4$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 1 \pm \sqrt{1 + 4 \cdot 4}}{2 \cdot - 1}$

Step 6.1.2.2

Multiply $4$ by $4$ .

$x = \frac{- 1 \pm \sqrt{1 + 16}}{2 \cdot - 1}$

Step 6.1.3

Add $1$ and $16$ .

$x = \frac{- 1 \pm \sqrt{17}}{2 \cdot - 1}$

Step 6.2

Multiply $2$ by $- 1$ .

$x = \frac{- 1 \pm \sqrt{17}}{- 2}$

Step 6.3

Simplify $\frac{- 1 \pm \sqrt{17}}{- 2}$ .

$x = \frac{1 \pm \sqrt{17}}{2}$

Step 6.4

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{17}}{2}$

Step 7

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{17}}{2}, \frac{1 - \sqrt{17}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 + \sqrt{17}}{2}, \frac{1 - \sqrt{17}}{2}$

Decimal Form:

$x = 2.56155281 \dots, - 1.56155281 \dots$