Calculus Examples

$0 = 4 x^{3} - 72 x^{2} + 324 x$

Step 1

Move all the expressions to the left side of the equation.

Tap for more steps...

Step 1.1

Subtract $4 x^{3}$ from both sides of the equation.

$- 4 x^{3} = - 72 x^{2} + 324 x$

Step 1.2

Add $72 x^{2}$ to both sides of the equation.

$- 4 x^{3} + 72 x^{2} = 324 x$

Step 1.3

Subtract $324 x$ from both sides of the equation.

$- 4 x^{3} + 72 x^{2} - 324 x = 0$

Step 2

Factor $4 x$ out of $- 4 x^{3} + 72 x^{2} - 324 x$ .

Tap for more steps...

Step 2.1

Factor $4 x$ out of $- 4 x^{3}$ .

$4 x (- x^{2}) + 72 x^{2} - 324 x = 0$

Step 2.2

Factor $4 x$ out of $72 x^{2}$ .

$4 x (- x^{2}) + 4 x (18 x) - 324 x = 0$

Step 2.3

Factor $4 x$ out of $- 324 x$ .

$4 x (- x^{2}) + 4 x (18 x) + 4 x (- 81) = 0$

Step 2.4

Factor $4 x$ out of $4 x (- x^{2}) + 4 x (18 x)$ .

$4 x (- x^{2} + 18 x) + 4 x (- 81) = 0$

Step 2.5

Factor $4 x$ out of $4 x (- x^{2} + 18 x) + 4 x (- 81)$ .

$4 x (- x^{2} + 18 x - 81) = 0$

Step 3

Factor.

Tap for more steps...

Step 3.1

Factor by grouping.

Tap for more steps...

Step 3.1.1

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 $a \cdot c = - 1 \cdot - 81 = 81$ and whose sum is $b = 18$ .

Tap for more steps...

Step 3.1.1.1

Factor $18$ out of $18 x$ .

$4 x (- x^{2} + 18 (x) - 81) = 0$

Step 3.1.1.2

Rewrite $18$ as $9$ plus $9$

$4 x (- x^{2} + (9 + 9) x - 81) = 0$

Step 3.1.1.3

Apply the distributive property.

$4 x (- x^{2} + 9 x + 9 x - 81) = 0$

Step 3.1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.1.2.1

Group the first two terms and the last two terms.

$4 x ((- x^{2} + 9 x) + 9 x - 81) = 0$

Step 3.1.2.2

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

$4 x (x (- x + 9) - 9 (- x + 9)) = 0$

Step 3.1.3

Factor the polynomial by factoring out the greatest common factor, $- x + 9$ .

$4 x ((- x + 9) (x - 9)) = 0$

Step 3.2

Remove unnecessary parentheses.

$4 x (- x + 9) (x - 9) = 0$

Step 4

Combine exponents.

Tap for more steps...

Step 4.1

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

$4 x (- (x) + 9) (x - 9) = 0$

Step 4.2

Rewrite $9$ as $- 1 (- 9)$ .

$4 x (- (x) - 1 \cdot - 9) (x - 9) = 0$

Step 4.3

Factor $- 1$ out of $- (x) - 1 (- 9)$ .

$4 x (- (x - 9)) (x - 9) = 0$

Step 4.4

Rewrite $- (x - 9)$ as $- 1 (x - 9)$ .

$4 x (- 1 (x - 9)) (x - 9) = 0$

Step 4.5

Remove parentheses.

$4 x \cdot (- 1 (x - 9) (x - 9)) = 0$

Step 4.6

Raise $x - 9$ to the power of $1$ .

$4 x \cdot (- 1 ((x - 9) (x - 9))) = 0$

Step 4.7

Raise $x - 9$ to the power of $1$ .

$4 x \cdot (- 1 ((x - 9) (x - 9))) = 0$

Step 4.8

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

$4 x \cdot (- 1 {(x - 9)}^{1 + 1}) = 0$

Step 4.9

Add $1$ and $1$ .

$4 x \cdot (- 1 {(x - 9)}^{2}) = 0$

Step 4.10

Multiply $- 1$ by $4$ .

$- 4 x {(x - 9)}^{2} = 0$

Step 5

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

$x = 0$

${(x - 9)}^{2} = 0$

Step 6

Set $x$ equal to $0$ .

$x = 0$

Step 7

Set ${(x - 9)}^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 7.1

Set ${(x - 9)}^{2}$ equal to $0$ .

${(x - 9)}^{2} = 0$

Step 7.2

Solve ${(x - 9)}^{2} = 0$ for $x$ .

Tap for more steps...

Step 7.2.1

Set the $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 7.2.2

Add $9$ to both sides of the equation.

$x = 9$

Step 8

The final solution is all the values that make $- 4 x {(x - 9)}^{2} = 0$ true.

$x = 0, 9$