Algebra Examples

$f (x) = - x^{3} + 15 x^{2} - 75 x + 125$

Step 1

Set $- x^{3} + 15 x^{2} - 75 x + 125$ equal to $0$ .

$- x^{3} + 15 x^{2} - 75 x + 125 = 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Factor the left side of the equation.

Tap for more steps...

Step 2.1.1

Regroup terms.

$- x^{3} + 125 + 15 x^{2} - 75 x = 0$

Step 2.1.2

Factor $- 1$ out of $- x^{3} + 125$ .

Tap for more steps...

Step 2.1.2.1

Factor $- 1$ out of $- x^{3}$ .

$- (x^{3}) + 125 + 15 x^{2} - 75 x = 0$

Step 2.1.2.2

Rewrite $125$ as $- 1 (- 125)$ .

$- (x^{3}) - 1 \cdot - 125 + 15 x^{2} - 75 x = 0$

Step 2.1.2.3

Factor $- 1$ out of $- (x^{3}) - 1 (- 125)$ .

$- (x^{3} - 125) + 15 x^{2} - 75 x = 0$

Step 2.1.3

Rewrite $125$ as $5^{3}$ .

$- (x^{3} - 5^{3}) + 15 x^{2} - 75 x = 0$

Step 2.1.4

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 5$ .

$- ((x - 5) (x^{2} + x \cdot 5 + 5^{2})) + 15 x^{2} - 75 x = 0$

Step 2.1.5

Factor.

Tap for more steps...

Step 2.1.5.1

Simplify.

Tap for more steps...

Step 2.1.5.1.1

Move $5$ to the left of $x$ .

$- ((x - 5) (x^{2} + 5 x + 5^{2})) + 15 x^{2} - 75 x = 0$

Step 2.1.5.1.2

Raise $5$ to the power of $2$ .

$- ((x - 5) (x^{2} + 5 x + 25)) + 15 x^{2} - 75 x = 0$

Step 2.1.5.2

Remove unnecessary parentheses.

$- (x - 5) (x^{2} + 5 x + 25) + 15 x^{2} - 75 x = 0$

Step 2.1.6

Factor $15 x$ out of $15 x^{2} - 75 x$ .

Tap for more steps...

Step 2.1.6.1

Factor $15 x$ out of $15 x^{2}$ .

$- (x - 5) (x^{2} + 5 x + 25) + 15 x (x) - 75 x = 0$

Step 2.1.6.2

Factor $15 x$ out of $- 75 x$ .

$- (x - 5) (x^{2} + 5 x + 25) + 15 x (x) + 15 x (- 5) = 0$

Step 2.1.6.3

Factor $15 x$ out of $15 x (x) + 15 x (- 5)$ .

$- (x - 5) (x^{2} + 5 x + 25) + 15 x (x - 5) = 0$

Step 2.1.7

Factor $x - 5$ out of $- (x - 5) (x^{2} + 5 x + 25) + 15 x (x - 5)$ .

Tap for more steps...

Step 2.1.7.1

Factor $x - 5$ out of $- (x - 5) (x^{2} + 5 x + 25)$ .

$(x - 5) (- 1 (x^{2} + 5 x + 25)) + 15 x (x - 5) = 0$

Step 2.1.7.2

Factor $x - 5$ out of $15 x (x - 5)$ .

$(x - 5) (- 1 (x^{2} + 5 x + 25)) + (x - 5) (15 x) = 0$

Step 2.1.7.3

Factor $x - 5$ out of $(x - 5) (- 1 (x^{2} + 5 x + 25)) + (x - 5) (15 x)$ .

$(x - 5) (- 1 (x^{2} + 5 x + 25) + 15 x) = 0$

Step 2.1.8

Apply the distributive property.

$(x - 5) (- 1 x^{2} - 1 (5 x) - 1 \cdot 25 + 15 x) = 0$

Step 2.1.9

Simplify.

Tap for more steps...

Step 2.1.9.1

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$(x - 5) (- x^{2} - 1 (5 x) - 1 \cdot 25 + 15 x) = 0$

Step 2.1.9.2

Multiply $5$ by $- 1$ .

$(x - 5) (- x^{2} - 5 x - 1 \cdot 25 + 15 x) = 0$

Step 2.1.9.3

Multiply $- 1$ by $25$ .

$(x - 5) (- x^{2} - 5 x - 25 + 15 x) = 0$

Step 2.1.10

Add $- 5 x$ and $15 x$ .

$(x - 5) (- x^{2} + 10 x - 25) = 0$

Step 2.1.11

Factor.

Tap for more steps...

Step 2.1.11.1

Factor by grouping.

Tap for more steps...

Step 2.1.11.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 25 = 25$ and whose sum is $b = 10$ .

Tap for more steps...

Step 2.1.11.1.1.1

Factor $10$ out of $10 x$ .

$(x - 5) (- x^{2} + 10 (x) - 25) = 0$

Step 2.1.11.1.1.2

Rewrite $10$ as $5$ plus $5$

$(x - 5) (- x^{2} + (5 + 5) x - 25) = 0$

Step 2.1.11.1.1.3

Apply the distributive property.

$(x - 5) (- x^{2} + 5 x + 5 x - 25) = 0$

Step 2.1.11.1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 2.1.11.1.2.1

Group the first two terms and the last two terms.

$(x - 5) ((- x^{2} + 5 x) + 5 x - 25) = 0$

Step 2.1.11.1.2.2

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

$(x - 5) (x (- x + 5) - 5 (- x + 5)) = 0$

Step 2.1.11.1.3

Factor the polynomial by factoring out the greatest common factor, $- x + 5$ .

$(x - 5) ((- x + 5) (x - 5)) = 0$

Step 2.1.11.2

Remove unnecessary parentheses.

$(x - 5) (- x + 5) (x - 5) = 0$

Step 2.1.12

Combine exponents.

Tap for more steps...

Step 2.1.12.1

Factor $- 1$ out of $x$ .

$(- 1 (- x) - 5) (- x + 5) (x - 5) = 0$

Step 2.1.12.2

Rewrite $- 5$ as $- 1 (5)$ .

$(- 1 (- x) - 1 \cdot 5) (- x + 5) (x - 5) = 0$

Step 2.1.12.3

Factor $- 1$ out of $- 1 (- x) - 1 (5)$ .

$- 1 (- x + 5) (- x + 5) (x - 5) = 0$

Step 2.1.12.4

Raise $- x + 5$ to the power of $1$ .

$- 1 ((- x + 5) (- x + 5)) (x - 5) = 0$

Step 2.1.12.5

Raise $- x + 5$ to the power of $1$ .

$- 1 ((- x + 5) (- x + 5)) (x - 5) = 0$

Step 2.1.12.6

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

$- 1 {(- x + 5)}^{1 + 1} (x - 5) = 0$

Step 2.1.12.7

Add $1$ and $1$ .

$- 1 {(- x + 5)}^{2} (x - 5) = 0$

Step 2.1.12.8

Factor $- 1$ out of $- x$ .

$- 1 {(- (x) + 5)}^{2} (x - 5) = 0$

Step 2.1.12.9

Rewrite $5$ as $- 1 (- 5)$ .

$- 1 {(- (x) - 1 \cdot - 5)}^{2} (x - 5) = 0$

Step 2.1.12.10

Factor $- 1$ out of $- (x) - 1 (- 5)$ .

$- 1 {(- (x - 5))}^{2} (x - 5) = 0$

Step 2.1.12.11

Rewrite $- (x - 5)$ as $- 1 (x - 5)$ .

$- 1 {(- 1 (x - 5))}^{2} (x - 5) = 0$

Step 2.1.12.12

Apply the product rule to $- 1 (x - 5)$ .

$- 1 ({(- 1)}^{2} {(x - 5)}^{2}) (x - 5) = 0$

Step 2.1.12.13

Raise $- 1$ to the power of $2$ .

$- 1 (1 {(x - 5)}^{2}) (x - 5) = 0$

Step 2.1.12.14

Multiply ${(x - 5)}^{2}$ by $1$ .

$- 1 {(x - 5)}^{2} (x - 5) = 0$

Step 2.1.12.15

Raise $x - 5$ to the power of $1$ .

$- 1 ((x - 5) {(x - 5)}^{2}) = 0$

Step 2.1.12.16

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

$- 1 {(x - 5)}^{1 + 2} = 0$

Step 2.1.12.17

Add $1$ and $2$ .

$- 1 {(x - 5)}^{3} = 0$

Step 2.2

Divide each term in $- 1 {(x - 5)}^{3} = 0$ by $- 1$ and simplify.

Tap for more steps...

Step 2.2.1

Divide each term in $- 1 {(x - 5)}^{3} = 0$ by $- 1$ .

$\frac{- 1 {(x - 5)}^{3}}{- 1} = \frac{0}{- 1}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{{(x - 5)}^{3}}{1} = \frac{0}{- 1}$

Step 2.2.2.2

Divide ${(x - 5)}^{3}$ by $1$ .

${(x - 5)}^{3} = \frac{0}{- 1}$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Divide $0$ by $- 1$ .

${(x - 5)}^{3} = 0$

Step 2.3

Set the $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 2.4

Add $5$ to both sides of the equation.

$x = 5$ (Multiplicity of $3$ )

Step 3