Algebra Examples

$(x - 2) (x + 1) = \frac{5 x + 21}{5}$

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 $(x - 2) (x + 1)$ .

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Step 1.1.1.1

Expand $(x - 2) (x + 1)$ using the FOIL Method.

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Step 1.1.1.1.1

Apply the distributive property.

$x (x + 1) - 2 (x + 1) = \frac{5 x + 21}{5}$

Step 1.1.1.1.2

Apply the distributive property.

$x \cdot x + x \cdot 1 - 2 (x + 1) = \frac{5 x + 21}{5}$

Step 1.1.1.1.3

Apply the distributive property.

$x \cdot x + x \cdot 1 - 2 x - 2 \cdot 1 = \frac{5 x + 21}{5}$

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 $x$ .

$x^{2} + x \cdot 1 - 2 x - 2 \cdot 1 = \frac{5 x + 21}{5}$

Step 1.1.1.2.1.2

Multiply $x$ by $1$ .

$x^{2} + x - 2 x - 2 \cdot 1 = \frac{5 x + 21}{5}$

Step 1.1.1.2.1.3

Multiply $- 2$ by $1$ .

$x^{2} + x - 2 x - 2 = \frac{5 x + 21}{5}$

Step 1.1.1.2.2

Subtract $2 x$ from $x$ .

$x^{2} - x - 2 = \frac{5 x + 21}{5}$

Step 1.2

Simplify the right side.

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Step 1.2.1

Simplify $\frac{5 x + 21}{5}$ .

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Step 1.2.1.1

Split the fraction $\frac{5 x + 21}{5}$ into two fractions.

$x^{2} - x - 2 = \frac{5 x}{5} + \frac{21}{5}$

Step 1.2.1.2

Cancel the common factor of $5$ .

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Step 1.2.1.2.1

Cancel the common factor.

$x^{2} - x - 2 = \frac{5 x}{5} + \frac{21}{5}$

Step 1.2.1.2.2

Divide $x$ by $1$ .

$x^{2} - x - 2 = x + \frac{21}{5}$

Step 1.3

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

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Step 1.3.1

Subtract $x$ from both sides of the equation.

$x^{2} - x - 2 - x = \frac{21}{5}$

Step 1.3.2

Subtract $\frac{21}{5}$ from both sides of the equation.

$x^{2} - x - 2 - x - \frac{21}{5} = 0$

Step 1.4

Simplify $x^{2} - x - 2 - x - \frac{21}{5}$ .

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Step 1.4.1

Subtract $x$ from $- x$ .

$x^{2} - 2 x - 2 - \frac{21}{5} = 0$

Step 1.4.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x^{2} - 2 x - 2 \cdot \frac{5}{5} - \frac{21}{5} = 0$

Step 1.4.3

Combine $- 2$ and $\frac{5}{5}$ .

$x^{2} - 2 x + \frac{- 2 \cdot 5}{5} - \frac{21}{5} = 0$

Step 1.4.4

Combine the numerators over the common denominator.

$x^{2} - 2 x + \frac{- 2 \cdot 5 - 21}{5} = 0$

Step 1.4.5

Simplify the numerator.

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Step 1.4.5.1

Multiply $- 2$ by $5$ .

$x^{2} - 2 x + \frac{- 10 - 21}{5} = 0$

Step 1.4.5.2

Subtract $21$ from $- 10$ .

$x^{2} - 2 x + \frac{- 31}{5} = 0$

Step 1.4.6

Move the negative in front of the fraction.

$x^{2} - 2 x - \frac{31}{5} = 0$

Step 2

Multiply through by the least common denominator $5$ , then simplify.

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Step 2.1

Apply the distributive property.

$5 x^{2} + 5 (- 2 x) + 5 (- \frac{31}{5}) = 0$

Step 2.2

Simplify.

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Step 2.2.1

Multiply $- 2$ by $5$ .

$5 x^{2} - 10 x + 5 (- \frac{31}{5}) = 0$

Step 2.2.2

Cancel the common factor of $5$ .

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Step 2.2.2.1

Move the leading negative in $- \frac{31}{5}$ into the numerator.

$5 x^{2} - 10 x + 5 (\frac{- 31}{5}) = 0$

Step 2.2.2.2

Cancel the common factor.

$5 x^{2} - 10 x + 5 (\frac{- 31}{5}) = 0$

Step 2.2.2.3

Rewrite the expression.

$5 x^{2} - 10 x - 31 = 0$

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 $a = 5$ , $b = - 10$ , and $c = - 31$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (5 \cdot - 31)}}{2 \cdot 5}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 5 \cdot - 31}}{2 \cdot 5}$

Step 5.1.2

Multiply $- 4 \cdot 5 \cdot - 31$ .

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Step 5.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{10 \pm \sqrt{100 - 20 \cdot - 31}}{2 \cdot 5}$

Step 5.1.2.2

Multiply $- 20$ by $- 31$ .

$x = \frac{10 \pm \sqrt{100 + 620}}{2 \cdot 5}$

Step 5.1.3

Add $100$ and $620$ .

$x = \frac{10 \pm \sqrt{720}}{2 \cdot 5}$

Step 5.1.4

Rewrite $720$ as $12^{2} \cdot 5$ .

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Step 5.1.4.1

Factor $144$ out of $720$ .

$x = \frac{10 \pm \sqrt{144 (5)}}{2 \cdot 5}$

Step 5.1.4.2

Rewrite $144$ as $12^{2}$ .

$x = \frac{10 \pm \sqrt{12^{2} \cdot 5}}{2 \cdot 5}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{10 \pm 12 \sqrt{5}}{2 \cdot 5}$

Step 5.2

Multiply $2$ by $5$ .

$x = \frac{10 \pm 12 \sqrt{5}}{10}$

Step 5.3

Simplify $\frac{10 \pm 12 \sqrt{5}}{10}$ .

$x = \frac{5 \pm 6 \sqrt{5}}{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \pm 6 \sqrt{5}}{5}$

Decimal Form:

$x = 3.68328157 \dots, - 1.68328157 \dots$