Algebra Examples

$133 = (3 x + 1) (x + 1)$

Step 1

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

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Step 1.1

Simplify the right side.

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Step 1.1.1

Simplify $(3 x + 1) (x + 1)$ .

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Step 1.1.1.1

Expand $(3 x + 1) (x + 1)$ using the FOIL Method.

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Step 1.1.1.1.1

Apply the distributive property.

$133 = 3 x (x + 1) + 1 (x + 1)$

Step 1.1.1.1.2

Apply the distributive property.

$133 = 3 x \cdot x + 3 x \cdot 1 + 1 (x + 1)$

Step 1.1.1.1.3

Apply the distributive property.

$133 = 3 x \cdot x + 3 x \cdot 1 + 1 x + 1 \cdot 1$

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 $x$ by adding the exponents.

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Step 1.1.1.2.1.1.1

Move $x$ .

$133 = 3 (x \cdot x) + 3 x \cdot 1 + 1 x + 1 \cdot 1$

Step 1.1.1.2.1.1.2

Multiply $x$ by $x$ .

$133 = 3 x^{2} + 3 x \cdot 1 + 1 x + 1 \cdot 1$

Step 1.1.1.2.1.2

Multiply $3$ by $1$ .

$133 = 3 x^{2} + 3 x + 1 x + 1 \cdot 1$

Step 1.1.1.2.1.3

Multiply $x$ by $1$ .

$133 = 3 x^{2} + 3 x + x + 1 \cdot 1$

Step 1.1.1.2.1.4

Multiply $1$ by $1$ .

$133 = 3 x^{2} + 3 x + x + 1$

Step 1.1.1.2.2

Add $3 x$ and $x$ .

$133 = 3 x^{2} + 4 x + 1$

Step 1.2

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

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Step 1.2.1

Subtract $3 x^{2}$ from both sides of the equation.

$133 - 3 x^{2} = 4 x + 1$

Step 1.2.2

Subtract $4 x$ from both sides of the equation.

$133 - 3 x^{2} - 4 x = 1$

Step 1.2.3

Subtract $1$ from both sides of the equation.

$133 - 3 x^{2} - 4 x - 1 = 0$

Step 1.3

Subtract $1$ from $133$ .

$- 3 x^{2} - 4 x + 132 = 0$

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 $a = - 3$ , $b = - 4$ , and $c = 132$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (- 3 \cdot 132)}}{2 \cdot - 3}$

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 $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 3 \cdot 132}}{2 \cdot - 3}$

Step 4.1.2

Multiply $- 4 \cdot - 3 \cdot 132$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{4 \pm \sqrt{16 + 12 \cdot 132}}{2 \cdot - 3}$

Step 4.1.2.2

Multiply $12$ by $132$ .

$x = \frac{4 \pm \sqrt{16 + 1584}}{2 \cdot - 3}$

Step 4.1.3

Add $16$ and $1584$ .

$x = \frac{4 \pm \sqrt{1600}}{2 \cdot - 3}$

Step 4.1.4

Rewrite $1600$ as $40^{2}$ .

$x = \frac{4 \pm \sqrt{40^{2}}}{2 \cdot - 3}$

Step 4.1.5

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

$x = \frac{4 \pm 40}{2 \cdot - 3}$

Step 4.2

Multiply $2$ by $- 3$ .

$x = \frac{4 \pm 40}{- 6}$

Step 4.3

Simplify $\frac{4 \pm 40}{- 6}$ .

$x = \frac{2 \pm 20}{- 3}$

Step 4.4

Move the negative in front of the fraction.

$x = - \frac{2 \pm 20}{3}$

Step 5

The final answer is the combination of both solutions.

$x = - \frac{22}{3}, 6$