Algebra Examples

$(x + 3) (x + 2) = x + 13$

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 $(x + 3) (x + 2)$ .

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Step 1.1.1.1

Expand $(x + 3) (x + 2)$ using the FOIL Method.

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Step 1.1.1.1.1

Apply the distributive property.

$x (x + 2) + 3 (x + 2) = x + 13$

Step 1.1.1.1.2

Apply the distributive property.

$x \cdot x + x \cdot 2 + 3 (x + 2) = x + 13$

Step 1.1.1.1.3

Apply the distributive property.

$x \cdot x + x \cdot 2 + 3 x + 3 \cdot 2 = x + 13$

Step 1.1.1.2

Simplify and combine like terms.

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Step 1.1.1.2.1

Simplify each term.

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Step 1.1.1.2.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 2 + 3 x + 3 \cdot 2 = x + 13$

Step 1.1.1.2.1.2

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

$x^{2} + 2 \cdot x + 3 x + 3 \cdot 2 = x + 13$

Step 1.1.1.2.1.3

Multiply $3$ by $2$ .

$x^{2} + 2 x + 3 x + 6 = x + 13$

Step 1.1.1.2.2

Add $2 x$ and $3 x$ .

$x^{2} + 5 x + 6 = x + 13$

Step 1.2

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

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Step 1.2.1

Subtract $x$ from both sides of the equation.

$x^{2} + 5 x + 6 - x = 13$

Step 1.2.2

Subtract $13$ from both sides of the equation.

$x^{2} + 5 x + 6 - x - 13 = 0$

Step 1.3

Simplify $x^{2} + 5 x + 6 - x - 13$ .

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Step 1.3.1

Subtract $x$ from $5 x$ .

$x^{2} + 4 x + 6 - 13 = 0$

Step 1.3.2

Subtract $13$ from $6$ .

$x^{2} + 4 x - 7 = 0$

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 $a = 1$ , $b = 4$ , and $c = - 7$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot - 7)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 7}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot - 7$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 7}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $- 7$ .

$x = \frac{- 4 \pm \sqrt{16 + 28}}{2 \cdot 1}$

Step 4.1.3

Add $16$ and $28$ .

$x = \frac{- 4 \pm \sqrt{44}}{2 \cdot 1}$

Step 4.1.4

Rewrite $44$ as $2^{2} \cdot 11$ .

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Step 4.1.4.1

Factor $4$ out of $44$ .

$x = \frac{- 4 \pm \sqrt{4 (11)}}{2 \cdot 1}$

Step 4.1.4.2

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 1}$

Step 4.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{11}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 \sqrt{11}}{2}$

Step 4.3

Simplify $\frac{- 4 \pm 2 \sqrt{11}}{2}$ .

$x = - 2 \pm \sqrt{11}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - 2 \pm \sqrt{11}$

Decimal Form:

$x = 1.31662479 \dots, - 5.31662479 \dots$