Algebra Examples

$\frac{1}{2} x (x - 7) (x + 9) = 0$

Step 1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot (x (x - 7) (x + 9))) = 2 \cdot 0$

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $2 (\frac{1}{2} \cdot (x (x - 7) (x + 9)))$ .

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Step 2.1.1.1

Simplify by multiplying through.

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Step 2.1.1.1.1

Apply the distributive property.

$2 (\frac{1}{2} \cdot ((x \cdot x + x \cdot - 7) (x + 9))) = 2 \cdot 0$

Step 2.1.1.1.2

Simplify the expression.

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Step 2.1.1.1.2.1

Multiply $x$ by $x$ .

$2 (\frac{1}{2} \cdot ((x^{2} + x \cdot - 7) (x + 9))) = 2 \cdot 0$

Step 2.1.1.1.2.2

Move $- 7$ to the left of $x$ .

$2 (\frac{1}{2} \cdot ((x^{2} - 7 \cdot x) (x + 9))) = 2 \cdot 0$

Step 2.1.1.2

Expand $(x^{2} - 7 x) (x + 9)$ using the FOIL Method.

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Step 2.1.1.2.1

Apply the distributive property.

$2 (\frac{1}{2} \cdot (x^{2} (x + 9) - 7 x (x + 9))) = 2 \cdot 0$

Step 2.1.1.2.2

Apply the distributive property.

$2 (\frac{1}{2} \cdot (x^{2} x + x^{2} \cdot 9 - 7 x (x + 9))) = 2 \cdot 0$

Step 2.1.1.2.3

Apply the distributive property.

$2 (\frac{1}{2} \cdot (x^{2} x + x^{2} \cdot 9 - 7 x \cdot x - 7 x \cdot 9)) = 2 \cdot 0$

Step 2.1.1.3

Simplify and combine like terms.

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Step 2.1.1.3.1

Simplify each term.

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Step 2.1.1.3.1.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.1.1.3.1.1.1

Multiply $x^{2}$ by $x$ .

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Step 2.1.1.3.1.1.1.1

Raise $x$ to the power of $1$ .

$2 (\frac{1}{2} \cdot (x^{2} x^{1} + x^{2} \cdot 9 - 7 x \cdot x - 7 x \cdot 9)) = 2 \cdot 0$

Step 2.1.1.3.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 (\frac{1}{2} \cdot (x^{2 + 1} + x^{2} \cdot 9 - 7 x \cdot x - 7 x \cdot 9)) = 2 \cdot 0$

Step 2.1.1.3.1.1.2

Add $2$ and $1$ .

$2 (\frac{1}{2} \cdot (x^{3} + x^{2} \cdot 9 - 7 x \cdot x - 7 x \cdot 9)) = 2 \cdot 0$

Step 2.1.1.3.1.2

Move $9$ to the left of $x^{2}$ .

$2 (\frac{1}{2} \cdot (x^{3} + 9 \cdot x^{2} - 7 x \cdot x - 7 x \cdot 9)) = 2 \cdot 0$

Step 2.1.1.3.1.3

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

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Step 2.1.1.3.1.3.1

Move $x$ .

$2 (\frac{1}{2} \cdot (x^{3} + 9 x^{2} - 7 (x \cdot x) - 7 x \cdot 9)) = 2 \cdot 0$

Step 2.1.1.3.1.3.2

Multiply $x$ by $x$ .

$2 (\frac{1}{2} \cdot (x^{3} + 9 x^{2} - 7 x^{2} - 7 x \cdot 9)) = 2 \cdot 0$

Step 2.1.1.3.1.4

Multiply $9$ by $- 7$ .

$2 (\frac{1}{2} \cdot (x^{3} + 9 x^{2} - 7 x^{2} - 63 x)) = 2 \cdot 0$

Step 2.1.1.3.2

Subtract $7 x^{2}$ from $9 x^{2}$ .

$2 (\frac{1}{2} \cdot (x^{3} + 2 x^{2} - 63 x)) = 2 \cdot 0$

Step 2.1.1.4

Apply the distributive property.

$2 (\frac{1}{2} x^{3} + \frac{1}{2} (2 x^{2}) + \frac{1}{2} (- 63 x)) = 2 \cdot 0$

Step 2.1.1.5

Simplify.

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Step 2.1.1.5.1

Combine $\frac{1}{2}$ and $x^{3}$ .

$2 (\frac{x^{3}}{2} + \frac{1}{2} (2 x^{2}) + \frac{1}{2} (- 63 x)) = 2 \cdot 0$

Step 2.1.1.5.2

Cancel the common factor of $2$ .

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Step 2.1.1.5.2.1

Factor $2$ out of $2 x^{2}$ .

$2 (\frac{x^{3}}{2} + \frac{1}{2} (2 (x^{2})) + \frac{1}{2} (- 63 x)) = 2 \cdot 0$

Step 2.1.1.5.2.2

Cancel the common factor.

$2 (\frac{x^{3}}{2} + \frac{1}{2} (2 x^{2}) + \frac{1}{2} (- 63 x)) = 2 \cdot 0$

Step 2.1.1.5.2.3

Rewrite the expression.

$2 (\frac{x^{3}}{2} + x^{2} + \frac{1}{2} (- 63 x)) = 2 \cdot 0$

Step 2.1.1.5.3

Multiply $\frac{1}{2} (- 63 x)$ .

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Step 2.1.1.5.3.1

Combine $- 63$ and $\frac{1}{2}$ .

$2 (\frac{x^{3}}{2} + x^{2} + \frac{- 63}{2} x) = 2 \cdot 0$

Step 2.1.1.5.3.2

Combine $\frac{- 63}{2}$ and $x$ .

$2 (\frac{x^{3}}{2} + x^{2} + \frac{- 63 x}{2}) = 2 \cdot 0$

Step 2.1.1.6

Move the negative in front of the fraction.

$2 (\frac{x^{3}}{2} + x^{2} - \frac{63 x}{2}) = 2 \cdot 0$

Step 2.1.1.7

Apply the distributive property.

$2 \frac{x^{3}}{2} + 2 x^{2} + 2 (- \frac{63 x}{2}) = 2 \cdot 0$

Step 2.1.1.8

Simplify.

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Step 2.1.1.8.1

Cancel the common factor of $2$ .

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Step 2.1.1.8.1.1

Cancel the common factor.

$2 \frac{x^{3}}{2} + 2 x^{2} + 2 (- \frac{63 x}{2}) = 2 \cdot 0$

Step 2.1.1.8.1.2

Rewrite the expression.

$x^{3} + 2 x^{2} + 2 (- \frac{63 x}{2}) = 2 \cdot 0$

Step 2.1.1.8.2

Cancel the common factor of $2$ .

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Step 2.1.1.8.2.1

Move the leading negative in $- \frac{63 x}{2}$ into the numerator.

$x^{3} + 2 x^{2} + 2 \frac{- 63 x}{2} = 2 \cdot 0$

Step 2.1.1.8.2.2

Cancel the common factor.

$x^{3} + 2 x^{2} + 2 \frac{- 63 x}{2} = 2 \cdot 0$

Step 2.1.1.8.2.3

Rewrite the expression.

$x^{3} + 2 x^{2} - 63 x = 2 \cdot 0$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $2$ by $0$ .

$x^{3} + 2 x^{2} - 63 x = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $x$ out of $x^{3} + 2 x^{2} - 63 x$ .

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Step 3.1.1

Factor $x$ out of $x^{3}$ .

$x \cdot x^{2} + 2 x^{2} - 63 x = 0$

Step 3.1.2

Factor $x$ out of $2 x^{2}$ .

$x \cdot x^{2} + x (2 x) - 63 x = 0$

Step 3.1.3

Factor $x$ out of $- 63 x$ .

$x \cdot x^{2} + x (2 x) + x \cdot - 63 = 0$

Step 3.1.4

Factor $x$ out of $x \cdot x^{2} + x (2 x)$ .

$x (x^{2} + 2 x) + x \cdot - 63 = 0$

Step 3.1.5

Factor $x$ out of $x (x^{2} + 2 x) + x \cdot - 63$ .

$x (x^{2} + 2 x - 63) = 0$

Step 3.2

Factor.

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Step 3.2.1

Factor $x^{2} + 2 x - 63$ using the AC method.

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Step 3.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 63$ and whose sum is $2$ .

$- 7, 9$

Step 3.2.1.2

Write the factored form using these integers.

$x ((x - 7) (x + 9)) = 0$

Step 3.2.2

Remove unnecessary parentheses.

$x (x - 7) (x + 9) = 0$

Step 4

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

$x = 0$

$x - 7 = 0$

$x + 9 = 0$

Step 5

Set $x$ equal to $0$ .

$x = 0$

Step 6

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

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Step 6.1

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

$x - 7 = 0$

Step 6.2

Add $7$ to both sides of the equation.

$x = 7$

Step 7

Set $x + 9$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Step 7.2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 8

The final solution is all the values that make $x (x - 7) (x + 9) = 0$ true.

$x = 0, 7, - 9$

Step 9