Basic Math Examples

$- a^{3} + 2 a^{2} + a - 2 = 0$

Step 1

Factor the left side of the equation.

Tap for more steps...

Step 1.1

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.1.1

Group the first two terms and the last two terms.

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

Step 1.1.2

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

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

Step 1.2

Factor the polynomial by factoring out the greatest common factor, $- a + 2$ .

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

Step 1.3

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

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

Step 1.4

Factor.

Tap for more steps...

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 $b = 1$ .

$(- a + 2) ((a + 1) (a - 1)) = 0$

Step 1.4.2

Remove unnecessary parentheses.

$(- a + 2) (a + 1) (a - 1) = 0$

Step 2

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

$- a + 2 = 0$

$a + 1 = 0$

$a - 1 = 0$

Step 3

Set $- a + 2$ equal to $0$ and solve for $a$ .

Tap for more steps...

Step 3.1

Set $- a + 2$ equal to $0$ .

$- a + 2 = 0$

Step 3.2

Solve $- a + 2 = 0$ for $a$ .

Tap for more steps...

Step 3.2.1

Subtract $2$ from both sides of the equation.

$- a = - 2$

Step 3.2.2

Divide each term in $- a = - 2$ by $- 1$ and simplify.

Tap for more steps...

Step 3.2.2.1

Divide each term in $- a = - 2$ by $- 1$ .

$\frac{- a}{- 1} = \frac{- 2}{- 1}$

Step 3.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{a}{1} = \frac{- 2}{- 1}$

Step 3.2.2.2.2

Divide $a$ by $1$ .

$a = \frac{- 2}{- 1}$

Step 3.2.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.2.3.1

Divide $- 2$ by $- 1$ .

$a = 2$

Step 4

Set $a + 1$ equal to $0$ and solve for $a$ .

Tap for more steps...

Step 4.1

Set $a + 1$ equal to $0$ .

$a + 1 = 0$

Step 4.2

Subtract $1$ from both sides of the equation.

$a = - 1$

Step 5

Set $a - 1$ equal to $0$ and solve for $a$ .

Tap for more steps...

Step 5.1

Set $a - 1$ equal to $0$ .

$a - 1 = 0$

Step 5.2

Add $1$ to both sides of the equation.

$a = 1$

Step 6

The final solution is all the values that make $(- a + 2) (a + 1) (a - 1) = 0$ true.

$a = 2, - 1, 1$