Finite Math Examples

$(4 x + 1) (x + 2) = - 2$

Step 1

Add $2$ to both sides of the equation.

$(4 x + 1) (x + 2) + 2 = 0$

Step 2

Simplify $(4 x + 1) (x + 2) + 2$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Expand $(4 x + 1) (x + 2)$ using the FOIL Method.

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Step 2.1.1.1

Apply the distributive property.

$4 x (x + 2) + 1 (x + 2) + 2 = 0$

Step 2.1.1.2

Apply the distributive property.

$4 x \cdot x + 4 x \cdot 2 + 1 (x + 2) + 2 = 0$

Step 2.1.1.3

Apply the distributive property.

$4 x \cdot x + 4 x \cdot 2 + 1 x + 1 \cdot 2 + 2 = 0$

Step 2.1.2

Simplify and combine like terms.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.2.1.1.1

Move $x$ .

$4 (x \cdot x) + 4 x \cdot 2 + 1 x + 1 \cdot 2 + 2 = 0$

Step 2.1.2.1.1.2

Multiply $x$ by $x$ .

$4 x^{2} + 4 x \cdot 2 + 1 x + 1 \cdot 2 + 2 = 0$

Step 2.1.2.1.2

Multiply $2$ by $4$ .

$4 x^{2} + 8 x + 1 x + 1 \cdot 2 + 2 = 0$

Step 2.1.2.1.3

Multiply $x$ by $1$ .

$4 x^{2} + 8 x + x + 1 \cdot 2 + 2 = 0$

Step 2.1.2.1.4

Multiply $2$ by $1$ .

$4 x^{2} + 8 x + x + 2 + 2 = 0$

Step 2.1.2.2

Add $8 x$ and $x$ .

$4 x^{2} + 9 x + 2 + 2 = 0$

Step 2.2

Add $2$ and $2$ .

$4 x^{2} + 9 x + 4 = 0$

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 $a = 4$ , $b = 9$ , and $c = 4$ into the quadratic formula and solve for $x$ .

$\frac{- 9 \pm \sqrt{9^{2} - 4 \cdot (4 \cdot 4)}}{2 \cdot 4}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 4 \cdot 4}}{2 \cdot 4}$

Step 5.1.2

Multiply $- 4 \cdot 4 \cdot 4$ .

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Step 5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 9 \pm \sqrt{81 - 16 \cdot 4}}{2 \cdot 4}$

Step 5.1.2.2

Multiply $- 16$ by $4$ .

$x = \frac{- 9 \pm \sqrt{81 - 64}}{2 \cdot 4}$

Step 5.1.3

Subtract $64$ from $81$ .

$x = \frac{- 9 \pm \sqrt{17}}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$x = \frac{- 9 \pm \sqrt{17}}{8}$

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

Raise $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 4 \cdot 4}}{2 \cdot 4}$

Step 6.1.2

Multiply $- 4 \cdot 4 \cdot 4$ .

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Step 6.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 9 \pm \sqrt{81 - 16 \cdot 4}}{2 \cdot 4}$

Step 6.1.2.2

Multiply $- 16$ by $4$ .

$x = \frac{- 9 \pm \sqrt{81 - 64}}{2 \cdot 4}$

Step 6.1.3

Subtract $64$ from $81$ .

$x = \frac{- 9 \pm \sqrt{17}}{2 \cdot 4}$

Step 6.2

Multiply $2$ by $4$ .

$x = \frac{- 9 \pm \sqrt{17}}{8}$

Step 6.3

Change the $\pm$ to $+$ .

$x = \frac{- 9 + \sqrt{17}}{8}$

Step 6.4

Rewrite $- 9$ as $- 1 (9)$ .

$x = \frac{- 1 \cdot 9 + \sqrt{17}}{8}$

Step 6.5

Factor $- 1$ out of $\sqrt{17}$ .

$x = \frac{- 1 \cdot 9 - 1 (- \sqrt{17})}{8}$

Step 6.6

Factor $- 1$ out of $- 1 (9) - 1 (- \sqrt{17})$ .

$x = \frac{- 1 (9 - \sqrt{17})}{8}$

Step 6.7

Move the negative in front of the fraction.

$x = - \frac{9 - \sqrt{17}}{8}$

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

Raise $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 4 \cdot 4}}{2 \cdot 4}$

Step 7.1.2

Multiply $- 4 \cdot 4 \cdot 4$ .

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Step 7.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 9 \pm \sqrt{81 - 16 \cdot 4}}{2 \cdot 4}$

Step 7.1.2.2

Multiply $- 16$ by $4$ .

$x = \frac{- 9 \pm \sqrt{81 - 64}}{2 \cdot 4}$

Step 7.1.3

Subtract $64$ from $81$ .

$x = \frac{- 9 \pm \sqrt{17}}{2 \cdot 4}$

Step 7.2

Multiply $2$ by $4$ .

$x = \frac{- 9 \pm \sqrt{17}}{8}$

Step 7.3

Change the $\pm$ to $-$ .

$x = \frac{- 9 - \sqrt{17}}{8}$

Step 7.4

Rewrite $- 9$ as $- 1 (9)$ .

$x = \frac{- 1 \cdot 9 - \sqrt{17}}{8}$

Step 7.5

Factor $- 1$ out of $- \sqrt{17}$ .

$x = \frac{- 1 \cdot 9 - (\sqrt{17})}{8}$

Step 7.6

Factor $- 1$ out of $- 1 (9) - (\sqrt{17})$ .

$x = \frac{- 1 (9 + \sqrt{17})}{8}$

Step 7.7

Move the negative in front of the fraction.

$x = - \frac{9 + \sqrt{17}}{8}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{9 - \sqrt{17}}{8}, - \frac{9 + \sqrt{17}}{8}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{9 - \sqrt{17}}{8}, - \frac{9 + \sqrt{17}}{8}$

Decimal Form:

$x = - 0.60961179 \dots, - 1.64038820 \dots$