Finite Math Examples

$4 x^{3} + x^{2} = 64 x + 16$

Step 1

Subtract $64 x$ from both sides of the equation.

$4 x^{3} + x^{2} - 64 x = 16$

Step 2

Subtract $16$ from both sides of the equation.

$4 x^{3} + x^{2} - 64 x - 16 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor out the greatest common factor from each group.

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Step 3.1.1

Group the first two terms and the last two terms.

$(4 x^{3} + x^{2}) - 64 x - 16 = 0$

Step 3.1.2

Factor out the greatest common factor (GCF) from each group.

$x^{2} (4 x + 1) - 16 (4 x + 1) = 0$

Step 3.2

Factor the polynomial by factoring out the greatest common factor, $4 x + 1$ .

$(4 x + 1) (x^{2} - 16) = 0$

Step 3.3

Rewrite $16$ as $4^{2}$ .

$(4 x + 1) (x^{2} - 4^{2}) = 0$

Step 3.4

Factor.

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Step 3.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

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

Step 3.4.2

Remove unnecessary parentheses.

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

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$4 x + 1 = 0$

$x + 4 = 0$

$x - 4 = 0$

Step 5

Set $4 x + 1$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $4 x + 1$ equal to $0$ .

$4 x + 1 = 0$

Step 5.2

Solve $4 x + 1 = 0$ for $x$ .

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Step 5.2.1

Subtract $1$ from both sides of the equation.

$4 x = - 1$

Step 5.2.2

Divide each term in $4 x = - 1$ by $4$ and simplify.

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Step 5.2.2.1

Divide each term in $4 x = - 1$ by $4$ .

$\frac{4 x}{4} = \frac{- 1}{4}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $4$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 1}{4}$

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{4}$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{4}$

Step 6

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 6.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 7

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 7.2

Add $4$ to both sides of the equation.

$x = 4$

Step 8

The final solution is all the values that make $(4 x + 1) (x + 4) (x - 4) = 0$ true.

$x = - \frac{1}{4}, - 4, 4$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{4}, - 4, 4$

Decimal Form:

$x = - 0.25, - 4, 4$