Algebra Examples

${(2 x - 7)}^{3} = {(2 x)}^{3} - 7^{3}$

Step 1

Simplify each term.

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Step 1.1

Apply the product rule to $2 x$ .

${(2 x - 7)}^{3} = 2^{3} x^{3} - 7^{3}$

Step 1.2

Raise $2$ to the power of $3$ .

${(2 x - 7)}^{3} = 8 x^{3} - 7^{3}$

Step 1.3

Raise $7$ to the power of $3$ .

${(2 x - 7)}^{3} = 8 x^{3} - 1 \cdot 343$

Step 1.4

Multiply $- 1$ by $343$ .

${(2 x - 7)}^{3} = 8 x^{3} - 343$

Step 2

Move all terms containing $x$ to the left side of the equation.

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Step 2.1

Subtract $8 x^{3}$ from both sides of the equation.

${(2 x - 7)}^{3} - 8 x^{3} = - 343$

Step 2.2

Simplify each term.

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Step 2.2.1

Use the Binomial Theorem.

${(2 x)}^{3} + 3 {(2 x)}^{2} \cdot - 7 + 3 (2 x) {(- 7)}^{2} + {(- 7)}^{3} - 8 x^{3} = - 343$

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Apply the product rule to $2 x$ .

$2^{3} x^{3} + 3 {(2 x)}^{2} \cdot - 7 + 3 (2 x) {(- 7)}^{2} + {(- 7)}^{3} - 8 x^{3} = - 343$

Step 2.2.2.2

Raise $2$ to the power of $3$ .

$8 x^{3} + 3 {(2 x)}^{2} \cdot - 7 + 3 (2 x) {(- 7)}^{2} + {(- 7)}^{3} - 8 x^{3} = - 343$

Step 2.2.2.3

Apply the product rule to $2 x$ .

$8 x^{3} + 3 (2^{2} x^{2}) \cdot - 7 + 3 (2 x) {(- 7)}^{2} + {(- 7)}^{3} - 8 x^{3} = - 343$

Step 2.2.2.4

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

$8 x^{3} + 3 (4 x^{2}) \cdot - 7 + 3 (2 x) {(- 7)}^{2} + {(- 7)}^{3} - 8 x^{3} = - 343$

Step 2.2.2.5

Multiply $4$ by $3$ .

$8 x^{3} + 12 x^{2} \cdot - 7 + 3 (2 x) {(- 7)}^{2} + {(- 7)}^{3} - 8 x^{3} = - 343$

Step 2.2.2.6

Multiply $- 7$ by $12$ .

$8 x^{3} - 84 x^{2} + 3 (2 x) {(- 7)}^{2} + {(- 7)}^{3} - 8 x^{3} = - 343$

Step 2.2.2.7

Multiply $2$ by $3$ .

$8 x^{3} - 84 x^{2} + 6 x {(- 7)}^{2} + {(- 7)}^{3} - 8 x^{3} = - 343$

Step 2.2.2.8

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

$8 x^{3} - 84 x^{2} + 6 x \cdot 49 + {(- 7)}^{3} - 8 x^{3} = - 343$

Step 2.2.2.9

Multiply $49$ by $6$ .

$8 x^{3} - 84 x^{2} + 294 x + {(- 7)}^{3} - 8 x^{3} = - 343$

Step 2.2.2.10

Raise $- 7$ to the power of $3$ .

$8 x^{3} - 84 x^{2} + 294 x - 343 - 8 x^{3} = - 343$

Step 2.3

Combine the opposite terms in $8 x^{3} - 84 x^{2} + 294 x - 343 - 8 x^{3}$ .

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Step 2.3.1

Subtract $8 x^{3}$ from $8 x^{3}$ .

$- 84 x^{2} + 294 x - 343 + 0 = - 343$

Step 2.3.2

Add $- 84 x^{2} + 294 x - 343$ and $0$ .

$- 84 x^{2} + 294 x - 343 = - 343$

Step 3

Add $343$ to both sides of the equation.

$- 84 x^{2} + 294 x - 343 + 343 = 0$

Step 4

Combine the opposite terms in $- 84 x^{2} + 294 x - 343 + 343$ .

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Step 4.1

Add $- 343$ and $343$ .

$- 84 x^{2} + 294 x + 0 = 0$

Step 4.2

Add $- 84 x^{2} + 294 x$ and $0$ .

$- 84 x^{2} + 294 x = 0$

Step 5

Factor $- 42 x$ out of $- 84 x^{2} + 294 x$ .

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Step 5.1

Factor $- 42 x$ out of $- 84 x^{2}$ .

$- 42 x (2 x) + 294 x = 0$

Step 5.2

Factor $- 42 x$ out of $294 x$ .

$- 42 x (2 x) - 42 x \cdot - 7 = 0$

Step 5.3

Factor $- 42 x$ out of $- 42 x (2 x) - 42 x (- 7)$ .

$- 42 x (2 x - 7) = 0$

Step 6

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

$x = 0$

$2 x - 7 = 0$

Step 7

Set $x$ equal to $0$ .

$x = 0$

Step 8

Set $2 x - 7$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $2 x - 7$ equal to $0$ .

$2 x - 7 = 0$

Step 8.2

Solve $2 x - 7 = 0$ for $x$ .

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Step 8.2.1

Add $7$ to both sides of the equation.

$2 x = 7$

Step 8.2.2

Divide each term in $2 x = 7$ by $2$ and simplify.

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Step 8.2.2.1

Divide each term in $2 x = 7$ by $2$ .

$\frac{2 x}{2} = \frac{7}{2}$

Step 8.2.2.2

Simplify the left side.

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Step 8.2.2.2.1

Cancel the common factor of $2$ .

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Step 8.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{7}{2}$

Step 8.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{2}$

Step 9

The final solution is all the values that make $- 42 x (2 x - 7) = 0$ true.

$x = 0, \frac{7}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{7}{2}$

Decimal Form:

$x = 0, 3.5$

Mixed Number Form:

$x = 0, 3 \frac{1}{2}$