Finite Math Examples

$64 w^{3} - 343 = 0$

Step 1

Add $343$ to both sides of the equation.

$64 w^{3} = 343$

Step 2

Subtract $343$ from both sides of the equation.

$64 w^{3} - 343 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Rewrite $64 w^{3}$ as ${(4 w)}^{3}$ .

${(4 w)}^{3} - 343 = 0$

Step 3.2

Rewrite $343$ as $7^{3}$ .

${(4 w)}^{3} - 7^{3} = 0$

Step 3.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 4 w$ and $b = 7$ .

$(4 w - 7) ({(4 w)}^{2} + 4 w \cdot 7 + 7^{2}) = 0$

Step 3.4

Simplify.

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Step 3.4.1

Apply the product rule to $4 w$ .

$(4 w - 7) (4^{2} w^{2} + 4 w \cdot 7 + 7^{2}) = 0$

Step 3.4.2

Raise $4$ to the power of $2$ .

$(4 w - 7) (16 w^{2} + 4 w \cdot 7 + 7^{2}) = 0$

Step 3.4.3

Multiply $7$ by $4$ .

$(4 w - 7) (16 w^{2} + 28 w + 7^{2}) = 0$

Step 3.4.4

Raise $7$ to the power of $2$ .

$(4 w - 7) (16 w^{2} + 28 w + 49) = 0$

Step 4

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

$4 w - 7 = 0$

$16 w^{2} + 28 w + 49 = 0$

Step 5

Set $4 w - 7$ equal to $0$ and solve for $w$ .

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Step 5.1

Set $4 w - 7$ equal to $0$ .

$4 w - 7 = 0$

Step 5.2

Solve $4 w - 7 = 0$ for $w$ .

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Step 5.2.1

Add $7$ to both sides of the equation.

$4 w = 7$

Step 5.2.2

Divide each term in $4 w = 7$ by $4$ and simplify.

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Step 5.2.2.1

Divide each term in $4 w = 7$ by $4$ .

$\frac{4 w}{4} = \frac{7}{4}$

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.

$\frac{4 w}{4} = \frac{7}{4}$

Step 5.2.2.2.1.2

Divide $w$ by $1$ .

$w = \frac{7}{4}$

Step 6

Set $16 w^{2} + 28 w + 49$ equal to $0$ and solve for $w$ .

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Step 6.1

Set $16 w^{2} + 28 w + 49$ equal to $0$ .

$16 w^{2} + 28 w + 49 = 0$

Step 6.2

Solve $16 w^{2} + 28 w + 49 = 0$ for $w$ .

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Step 6.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6.2.2

Substitute the values $a = 16$ , $b = 28$ , and $c = 49$ into the quadratic formula and solve for $w$ .

$\frac{- 28 \pm \sqrt{28^{2} - 4 \cdot (16 \cdot 49)}}{2 \cdot 16}$

Step 6.2.3

Simplify.

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Step 6.2.3.1

Simplify the numerator.

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Step 6.2.3.1.1

Raise $28$ to the power of $2$ .

$w = \frac{- 28 \pm \sqrt{784 - 4 \cdot 16 \cdot 49}}{2 \cdot 16}$

Step 6.2.3.1.2

Multiply $- 4 \cdot 16 \cdot 49$ .

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Step 6.2.3.1.2.1

Multiply $- 4$ by $16$ .

$w = \frac{- 28 \pm \sqrt{784 - 64 \cdot 49}}{2 \cdot 16}$

Step 6.2.3.1.2.2

Multiply $- 64$ by $49$ .

$w = \frac{- 28 \pm \sqrt{784 - 3136}}{2 \cdot 16}$

Step 6.2.3.1.3

Subtract $3136$ from $784$ .

$w = \frac{- 28 \pm \sqrt{- 2352}}{2 \cdot 16}$

Step 6.2.3.1.4

Rewrite $- 2352$ as $- 1 (2352)$ .

$w = \frac{- 28 \pm \sqrt{- 1 \cdot 2352}}{2 \cdot 16}$

Step 6.2.3.1.5

Rewrite $\sqrt{- 1 (2352)}$ as $\sqrt{- 1} \cdot \sqrt{2352}$ .

$w = \frac{- 28 \pm \sqrt{- 1} \cdot \sqrt{2352}}{2 \cdot 16}$

Step 6.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$w = \frac{- 28 \pm i \cdot \sqrt{2352}}{2 \cdot 16}$

Step 6.2.3.1.7

Rewrite $2352$ as $28^{2} \cdot 3$ .

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Step 6.2.3.1.7.1

Factor $784$ out of $2352$ .

$w = \frac{- 28 \pm i \cdot \sqrt{784 (3)}}{2 \cdot 16}$

Step 6.2.3.1.7.2

Rewrite $784$ as $28^{2}$ .

$w = \frac{- 28 \pm i \cdot \sqrt{28^{2} \cdot 3}}{2 \cdot 16}$

Step 6.2.3.1.8

Pull terms out from under the radical.

$w = \frac{- 28 \pm i \cdot (28 \sqrt{3})}{2 \cdot 16}$

Step 6.2.3.1.9

Move $28$ to the left of $i$ .

$w = \frac{- 28 \pm 28 i \sqrt{3}}{2 \cdot 16}$

Step 6.2.3.2

Multiply $2$ by $16$ .

$w = \frac{- 28 \pm 28 i \sqrt{3}}{32}$

Step 6.2.3.3

Simplify $\frac{- 28 \pm 28 i \sqrt{3}}{32}$ .

$w = \frac{- 7 \pm 7 i \sqrt{3}}{8}$

Step 6.2.4

The final answer is the combination of both solutions.

$w = - \frac{7 - 7 i \sqrt{3}}{8}, - \frac{7 + 7 i \sqrt{3}}{8}$

Step 7

The final solution is all the values that make $(4 w - 7) (16 w^{2} + 28 w + 49) = 0$ true.

$w = \frac{7}{4}, - \frac{7 - 7 i \sqrt{3}}{8}, - \frac{7 + 7 i \sqrt{3}}{8}$

Step 8