Finite Math Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Apply the distributive property.

$x (4 x) + x \cdot 1 + 4 (4 x + 1) = - 1$

Step 1.1.3

Apply the distributive property.

$x (4 x) + x \cdot 1 + 4 (4 x) + 4 \cdot 1 = - 1$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Rewrite using the commutative property of multiplication.

$4 x \cdot x + x \cdot 1 + 4 (4 x) + 4 \cdot 1 = - 1$

Step 1.2.1.2

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

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Step 1.2.1.2.1

Move $x$ .

$4 (x \cdot x) + x \cdot 1 + 4 (4 x) + 4 \cdot 1 = - 1$

Step 1.2.1.2.2

Multiply $x$ by $x$ .

$4 x^{2} + x \cdot 1 + 4 (4 x) + 4 \cdot 1 = - 1$

Step 1.2.1.3

Multiply $x$ by $1$ .

$4 x^{2} + x + 4 (4 x) + 4 \cdot 1 = - 1$

Step 1.2.1.4

Multiply $4$ by $4$ .

$4 x^{2} + x + 16 x + 4 \cdot 1 = - 1$

Step 1.2.1.5

Multiply $4$ by $1$ .

$4 x^{2} + x + 16 x + 4 = - 1$

Step 1.2.2

Add $x$ and $16 x$ .

$4 x^{2} + 17 x + 4 = - 1$

Step 2

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

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Step 2.1

Add $1$ to both sides of the equation.

$4 x^{2} + 17 x + 4 + 1 = 0$

Step 2.2

Add $4$ and $1$ .

$4 x^{2} + 17 x + 5 = 0$

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 $a = 4$ , $b = 17$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{- 17 \pm \sqrt{17^{2} - 4 \cdot (4 \cdot 5)}}{2 \cdot 4}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $17$ to the power of $2$ .

$x = \frac{- 17 \pm \sqrt{289 - 4 \cdot 4 \cdot 5}}{2 \cdot 4}$

Step 5.1.2

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

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Step 5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 17 \pm \sqrt{289 - 16 \cdot 5}}{2 \cdot 4}$

Step 5.1.2.2

Multiply $- 16$ by $5$ .

$x = \frac{- 17 \pm \sqrt{289 - 80}}{2 \cdot 4}$

Step 5.1.3

Subtract $80$ from $289$ .

$x = \frac{- 17 \pm \sqrt{209}}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$x = \frac{- 17 \pm \sqrt{209}}{8}$

Step 6

The final answer is the combination of both solutions.

$x = - \frac{17 - \sqrt{209}}{8}, - \frac{17 + \sqrt{209}}{8}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{17 - \sqrt{209}}{8}, - \frac{17 + \sqrt{209}}{8}$

Decimal Form:

$x = - 0.31789596 \dots, - 3.93210403 \dots$