Basic Math Examples

$128 a^{3} + 686 = 0$

Step 1

Subtract $686$ from both sides of the equation.

$128 a^{3} = - 686$

Step 2

Add $686$ to both sides of the equation.

$128 a^{3} + 686 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $2$ out of $128 a^{3} + 686$ .

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Step 3.1.1

Factor $2$ out of $128 a^{3}$ .

$2 (64 a^{3}) + 686 = 0$

Step 3.1.2

Factor $2$ out of $686$ .

$2 (64 a^{3}) + 2 (343) = 0$

Step 3.1.3

Factor $2$ out of $2 (64 a^{3}) + 2 (343)$ .

$2 (64 a^{3} + 343) = 0$

Step 3.2

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

$2 ({(4 a)}^{3} + 343) = 0$

Step 3.3

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

$2 ({(4 a)}^{3} + 7^{3}) = 0$

Step 3.4

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

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

Step 3.5

Factor.

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Step 3.5.1

Simplify.

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Step 3.5.1.1

Apply the product rule to $4 a$ .

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

Step 3.5.1.2

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

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

Step 3.5.1.3

Multiply $4$ by $- 1$ .

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

Step 3.5.1.4

Multiply $7$ by $- 4$ .

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

Step 3.5.1.5

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

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

Step 3.5.2

Remove unnecessary parentheses.

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

Step 4

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

$4 a + 7 = 0$

$16 a^{2} - 28 a + 49 = 0$

Step 5

Set $4 a + 7$ equal to $0$ and solve for $a$ .

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Step 5.1

Set $4 a + 7$ equal to $0$ .

$4 a + 7 = 0$

Step 5.2

Solve $4 a + 7 = 0$ for $a$ .

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Step 5.2.1

Subtract $7$ from both sides of the equation.

$4 a = - 7$

Step 5.2.2

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

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Step 5.2.2.1

Divide each term in $4 a = - 7$ by $4$ .

$\frac{4 a}{4} = \frac{- 7}{4}$

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.

$\frac{4 a}{4} = \frac{- 7}{4}$

Step 5.2.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 7}{4}$

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.

$a = - \frac{7}{4}$

Step 6

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

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Step 6.1

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

$16 a^{2} - 28 a + 49 = 0$

Step 6.2

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

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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 $a$ .

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

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 $2$ .

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

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 $16$ .

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

Step 6.2.3.1.2.2

Multiply $- 64$ by $49$ .

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

Step 6.2.3.1.3

Subtract $3136$ from $784$ .

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

Step 6.2.3.1.4

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

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

Step 6.2.3.1.5

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

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

Step 6.2.3.1.6

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

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

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 $2352$ .

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

Step 6.2.3.1.7.2

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

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

Step 6.2.3.1.8

Pull terms out from under the radical.

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

Step 6.2.3.1.9

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

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

Step 6.2.3.2

Multiply $2$ by $16$ .

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

Step 6.2.3.3

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

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

Step 6.2.4

The final answer is the combination of both solutions.

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

Step 7

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

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