Basic Math Examples

$c^{3} + 3 c^{2} = 25 c + 75$

Step 1

Subtract $25 c$ from both sides of the equation.

$c^{3} + 3 c^{2} - 25 c = 75$

Step 2

Subtract $75$ from both sides of the equation.

$c^{3} + 3 c^{2} - 25 c - 75 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor out the greatest common factor from each group.

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Step 3.1.1

Group the first two terms and the last two terms.

$(c^{3} + 3 c^{2}) - 25 c - 75 = 0$

Step 3.1.2

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

$c^{2} (c + 3) - 25 (c + 3) = 0$

Step 3.2

Factor the polynomial by factoring out the greatest common factor, $c + 3$ .

$(c + 3) (c^{2} - 25) = 0$

Step 3.3

Rewrite $25$ as $5^{2}$ .

$(c + 3) (c^{2} - 5^{2}) = 0$

Step 3.4

Factor.

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Step 3.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = c$ and $b = 5$ .

$(c + 3) ((c + 5) (c - 5)) = 0$

Step 3.4.2

Remove unnecessary parentheses.

$(c + 3) (c + 5) (c - 5) = 0$

Step 4

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

$c + 3 = 0$

$c + 5 = 0$

$c - 5 = 0$

Step 5

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

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Step 5.1

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

$c + 3 = 0$

Step 5.2

Subtract $3$ from both sides of the equation.

$c = - 3$

Step 6

Set $c + 5$ equal to $0$ and solve for $c$ .

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Step 6.1

Set $c + 5$ equal to $0$ .

$c + 5 = 0$

Step 6.2

Subtract $5$ from both sides of the equation.

$c = - 5$

Step 7

Set $c - 5$ equal to $0$ and solve for $c$ .

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Step 7.1

Set $c - 5$ equal to $0$ .

$c - 5 = 0$

Step 7.2

Add $5$ to both sides of the equation.

$c = 5$

Step 8

The final solution is all the values that make $(c + 3) (c + 5) (c - 5) = 0$ true.

$c = - 3, - 5, 5$