Pre-Algebra Examples

$x \cdot (x + 1) = 399$

Step 1

Simplify $x \cdot (x + 1)$ .

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Step 1.1

Apply the distributive property.

$x \cdot x + x \cdot 1 = 399$

Step 1.2

Simplify the expression.

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Step 1.2.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 1 = 399$

Step 1.2.2

Multiply $x$ by $1$ .

$x^{2} + x = 399$

Step 2

Subtract $399$ from both sides of the equation.

$x^{2} + x - 399 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 1$ , $b = 1$ , and $c = - 399$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot - 399)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 399}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot - 399$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 399}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $- 399$ .

$x = \frac{- 1 \pm \sqrt{1 + 1596}}{2 \cdot 1}$

Step 5.1.3

Add $1$ and $1596$ .

$x = \frac{- 1 \pm \sqrt{1597}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1597}}{2}$

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.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 399}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot - 399$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 399}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $- 399$ .

$x = \frac{- 1 \pm \sqrt{1 + 1596}}{2 \cdot 1}$

Step 6.1.3

Add $1$ and $1596$ .

$x = \frac{- 1 \pm \sqrt{1597}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1597}}{2}$

Step 6.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + \sqrt{1597}}{2}$

Step 6.4

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

$x = \frac{- 1 \cdot 1 + \sqrt{1597}}{2}$

Step 6.5

Factor $- 1$ out of $\sqrt{1597}$ .

$x = \frac{- 1 \cdot 1 - 1 (- \sqrt{1597})}{2}$

Step 6.6

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{1597})$ .

$x = \frac{- 1 (1 - \sqrt{1597})}{2}$

Step 6.7

Move the negative in front of the fraction.

$x = - \frac{1 - \sqrt{1597}}{2}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 399}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot - 399$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 399}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $- 399$ .

$x = \frac{- 1 \pm \sqrt{1 + 1596}}{2 \cdot 1}$

Step 7.1.3

Add $1$ and $1596$ .

$x = \frac{- 1 \pm \sqrt{1597}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1597}}{2}$

Step 7.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - \sqrt{1597}}{2}$

Step 7.4

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

$x = \frac{- 1 \cdot 1 - \sqrt{1597}}{2}$

Step 7.5

Factor $- 1$ out of $- \sqrt{1597}$ .

$x = \frac{- 1 \cdot 1 - (\sqrt{1597})}{2}$

Step 7.6

Factor $- 1$ out of $- 1 (1) - (\sqrt{1597})$ .

$x = \frac{- 1 (1 + \sqrt{1597})}{2}$

Step 7.7

Move the negative in front of the fraction.

$x = - \frac{1 + \sqrt{1597}}{2}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{1597}}{2}, - \frac{1 + \sqrt{1597}}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1 - \sqrt{1597}}{2}, - \frac{1 + \sqrt{1597}}{2}$

Decimal Form:

$x = 19.48124120 \dots, - 20.48124120 \dots$