Algebra Examples

$x (x + 1) = x + (x + 1) + 19$

Step 1

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

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify $x (x + 1)$ .

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Step 1.1.1.1

Remove parentheses.

$x (x + 1) = x + x + 1 + 19$

Step 1.1.1.2

Apply the distributive property.

$x \cdot x + x \cdot 1 = x + x + 1 + 19$

Step 1.1.1.3

Simplify the expression.

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Step 1.1.1.3.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 1 = x + x + 1 + 19$

Step 1.1.1.3.2

Multiply $x$ by $1$ .

$x^{2} + x = x + x + 1 + 19$

Step 1.2

Simplify the right side.

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Step 1.2.1

Simplify $x + x + 1 + 19$ .

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Step 1.2.1.1

Add $x$ and $x$ .

$x^{2} + x = 2 x + 1 + 19$

Step 1.2.1.2

Add $1$ and $19$ .

$x^{2} + x = 2 x + 20$

Step 1.3

Move all the expressions to the left side of the equation.

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Step 1.3.1

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

$x^{2} + x - 2 x = 20$

Step 1.3.2

Subtract $20$ from both sides of the equation.

$x^{2} + x - 2 x - 20 = 0$

Step 1.4

Subtract $2 x$ from $x$ .

$x^{2} - x - 20 = 0$

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 $c = - 20$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot - 20)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 20}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot - 20$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot - 20}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $- 20$ .

$x = \frac{1 \pm \sqrt{1 + 80}}{2 \cdot 1}$

Step 4.1.3

Add $1$ and $80$ .

$x = \frac{1 \pm \sqrt{81}}{2 \cdot 1}$

Step 4.1.4

Rewrite $81$ as $9^{2}$ .

$x = \frac{1 \pm \sqrt{9^{2}}}{2 \cdot 1}$

Step 4.1.5

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

$x = \frac{1 \pm 9}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm 9}{2}$

Step 5

The final answer is the combination of both solutions.

$x = 5, - 4$