Basic Math Examples

$a^{3} - 11 a^{2} - 9 a + 99 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor out the greatest common factor from each group.

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Step 1.1.1

Group the first two terms and the last two terms.

$(a^{3} - 11 a^{2}) - 9 a + 99 = 0$

Step 1.1.2

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

$a^{2} (a - 11) - 9 (a - 11) = 0$

Step 1.2

Factor the polynomial by factoring out the greatest common factor, $a - 11$ .

$(a - 11) (a^{2} - 9) = 0$

Step 1.3

Rewrite $9$ as $3^{2}$ .

$(a - 11) (a^{2} - 3^{2}) = 0$

Step 1.4

Factor.

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Step 1.4.1

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

$(a - 11) ((a + 3) (a - 3)) = 0$

Step 1.4.2

Remove unnecessary parentheses.

$(a - 11) (a + 3) (a - 3) = 0$

Step 2

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

$a - 11 = 0$

$a + 3 = 0$

$a - 3 = 0$

Step 3

Set $a - 11$ equal to $0$ and solve for $a$ .

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Step 3.1

Set $a - 11$ equal to $0$ .

$a - 11 = 0$

Step 3.2

Add $11$ to both sides of the equation.

$a = 11$

Step 4

Set $a + 3$ equal to $0$ and solve for $a$ .

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Step 4.1

Set $a + 3$ equal to $0$ .

$a + 3 = 0$

Step 4.2

Subtract $3$ from both sides of the equation.

$a = - 3$

Step 5

Set $a - 3$ equal to $0$ and solve for $a$ .

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Step 5.1

Set $a - 3$ equal to $0$ .

$a - 3 = 0$

Step 5.2

Add $3$ to both sides of the equation.

$a = 3$

Step 6

The final solution is all the values that make $(a - 11) (a + 3) (a - 3) = 0$ true.

$a = 11, - 3, 3$