Algebra Examples

$6 x^{3} + 4 x^{2} - 6 x - 4 = 2 (x + k) (x - k) (3 x + 2)$

Step 1

Simplify $2 (x + k) (x - k) (3 x + 2)$ .

Tap for more steps...

Step 1.1

Rewrite.

$6 x^{3} + 4 x^{2} - 6 x - 4 = 0 + 0 + 2 (x + k) (x - k) (3 x + 2)$

Step 1.2

Apply the distributive property.

$6 x^{3} + 4 x^{2} - 6 x - 4 = (2 x + 2 k) (x - k) (3 x + 2)$

Step 1.3

Expand $(2 x + 2 k) (x - k)$ using the FOIL Method.

Tap for more steps...

Step 1.3.1

Apply the distributive property.

$6 x^{3} + 4 x^{2} - 6 x - 4 = (2 x (x - k) + 2 k (x - k)) (3 x + 2)$

Step 1.3.2

Apply the distributive property.

$6 x^{3} + 4 x^{2} - 6 x - 4 = (2 x \cdot x + 2 x (- k) + 2 k (x - k)) (3 x + 2)$

Step 1.3.3

Apply the distributive property.

$6 x^{3} + 4 x^{2} - 6 x - 4 = (2 x \cdot x + 2 x (- k) + 2 k x + 2 k (- k)) (3 x + 2)$

Step 1.4

Simplify and combine like terms.

Tap for more steps...

Step 1.4.1

Simplify each term.

Tap for more steps...

Step 1.4.1.1

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.4.1.1.1

Move $x$ .

$6 x^{3} + 4 x^{2} - 6 x - 4 = (2 (x \cdot x) + 2 x (- k) + 2 k x + 2 k (- k)) (3 x + 2)$

Step 1.4.1.1.2

Multiply $x$ by $x$ .

$6 x^{3} + 4 x^{2} - 6 x - 4 = (2 x^{2} + 2 x (- k) + 2 k x + 2 k (- k)) (3 x + 2)$

Step 1.4.1.2

Rewrite using the commutative property of multiplication.

$6 x^{3} + 4 x^{2} - 6 x - 4 = (2 x^{2} + 2 \cdot - 1 x k + 2 k x + 2 k (- k)) (3 x + 2)$

Step 1.4.1.3

Multiply $2$ by $- 1$ .

$6 x^{3} + 4 x^{2} - 6 x - 4 = (2 x^{2} - 2 x k + 2 k x + 2 k (- k)) (3 x + 2)$

Step 1.4.1.4

Rewrite using the commutative property of multiplication.

$6 x^{3} + 4 x^{2} - 6 x - 4 = (2 x^{2} - 2 x k + 2 k x + 2 \cdot - 1 k \cdot k) (3 x + 2)$

Step 1.4.1.5

Multiply $k$ by $k$ by adding the exponents.

Tap for more steps...

Step 1.4.1.5.1

Move $k$ .

$6 x^{3} + 4 x^{2} - 6 x - 4 = (2 x^{2} - 2 x k + 2 k x + 2 \cdot - 1 (k \cdot k)) (3 x + 2)$

Step 1.4.1.5.2

Multiply $k$ by $k$ .

$6 x^{3} + 4 x^{2} - 6 x - 4 = (2 x^{2} - 2 x k + 2 k x + 2 \cdot - 1 k^{2}) (3 x + 2)$

Step 1.4.1.6

Multiply $2$ by $- 1$ .

$6 x^{3} + 4 x^{2} - 6 x - 4 = (2 x^{2} - 2 x k + 2 k x - 2 k^{2}) (3 x + 2)$

Step 1.4.2

Add $- 2 x k$ and $2 k x$ .

Tap for more steps...

Step 1.4.2.1

Move $x$ .

$6 x^{3} + 4 x^{2} - 6 x - 4 = (2 x^{2} - 2 k x + 2 k x - 2 k^{2}) (3 x + 2)$

Step 1.4.2.2

Add $- 2 k x$ and $2 k x$ .

$6 x^{3} + 4 x^{2} - 6 x - 4 = (2 x^{2} + 0 - 2 k^{2}) (3 x + 2)$

Step 1.4.3

Add $2 x^{2}$ and $0$ .

$6 x^{3} + 4 x^{2} - 6 x - 4 = (2 x^{2} - 2 k^{2}) (3 x + 2)$

Step 1.5

Expand $(2 x^{2} - 2 k^{2}) (3 x + 2)$ using the FOIL Method.

Tap for more steps...

Step 1.5.1

Apply the distributive property.

$6 x^{3} + 4 x^{2} - 6 x - 4 = 2 x^{2} (3 x + 2) - 2 k^{2} (3 x + 2)$

Step 1.5.2

Apply the distributive property.

$6 x^{3} + 4 x^{2} - 6 x - 4 = 2 x^{2} (3 x) + 2 x^{2} \cdot 2 - 2 k^{2} (3 x + 2)$

Step 1.5.3

Apply the distributive property.

$6 x^{3} + 4 x^{2} - 6 x - 4 = 2 x^{2} (3 x) + 2 x^{2} \cdot 2 - 2 k^{2} (3 x) - 2 k^{2} \cdot 2$

Step 1.6

Simplify each term.

Tap for more steps...

Step 1.6.1

Rewrite using the commutative property of multiplication.

$6 x^{3} + 4 x^{2} - 6 x - 4 = 2 \cdot 3 x^{2} x + 2 x^{2} \cdot 2 - 2 k^{2} (3 x) - 2 k^{2} \cdot 2$

Step 1.6.2

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.6.2.1

Move $x$ .

$6 x^{3} + 4 x^{2} - 6 x - 4 = 2 \cdot 3 (x \cdot x^{2}) + 2 x^{2} \cdot 2 - 2 k^{2} (3 x) - 2 k^{2} \cdot 2$

Step 1.6.2.2

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 1.6.2.2.1

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

$6 x^{3} + 4 x^{2} - 6 x - 4 = 2 \cdot 3 (x^{1} x^{2}) + 2 x^{2} \cdot 2 - 2 k^{2} (3 x) - 2 k^{2} \cdot 2$

Step 1.6.2.2.2

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

$6 x^{3} + 4 x^{2} - 6 x - 4 = 2 \cdot 3 x^{1 + 2} + 2 x^{2} \cdot 2 - 2 k^{2} (3 x) - 2 k^{2} \cdot 2$

Step 1.6.2.3

Add $1$ and $2$ .

$6 x^{3} + 4 x^{2} - 6 x - 4 = 2 \cdot 3 x^{3} + 2 x^{2} \cdot 2 - 2 k^{2} (3 x) - 2 k^{2} \cdot 2$

Step 1.6.3

Multiply $2$ by $3$ .

$6 x^{3} + 4 x^{2} - 6 x - 4 = 6 x^{3} + 2 x^{2} \cdot 2 - 2 k^{2} (3 x) - 2 k^{2} \cdot 2$

Step 1.6.4

Multiply $2$ by $2$ .

$6 x^{3} + 4 x^{2} - 6 x - 4 = 6 x^{3} + 4 x^{2} - 2 k^{2} (3 x) - 2 k^{2} \cdot 2$

Step 1.6.5

Rewrite using the commutative property of multiplication.

$6 x^{3} + 4 x^{2} - 6 x - 4 = 6 x^{3} + 4 x^{2} - 2 \cdot 3 k^{2} x - 2 k^{2} \cdot 2$

Step 1.6.6

Multiply $- 2$ by $3$ .

$6 x^{3} + 4 x^{2} - 6 x - 4 = 6 x^{3} + 4 x^{2} - 6 k^{2} x - 2 k^{2} \cdot 2$

Step 1.6.7

Multiply $2$ by $- 2$ .

$6 x^{3} + 4 x^{2} - 6 x - 4 = 6 x^{3} + 4 x^{2} - 6 k^{2} x - 4 k^{2}$

Step 2

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 2.1

Subtract $6 x^{3}$ from both sides of the equation.

$6 x^{3} + 4 x^{2} - 6 x - 4 - 6 x^{3} = 4 x^{2} - 6 k^{2} x - 4 k^{2}$

Step 2.2

Subtract $4 x^{2}$ from both sides of the equation.

$6 x^{3} + 4 x^{2} - 6 x - 4 - 6 x^{3} - 4 x^{2} = - 6 k^{2} x - 4 k^{2}$

Step 2.3

Add $6 k^{2} x$ to both sides of the equation.

$6 x^{3} + 4 x^{2} - 6 x - 4 - 6 x^{3} - 4 x^{2} + 6 k^{2} x = - 4 k^{2}$

Step 2.4

Combine the opposite terms in $6 x^{3} + 4 x^{2} - 6 x - 4 - 6 x^{3} - 4 x^{2} + 6 k^{2} x$ .

Tap for more steps...

Step 2.4.1

Subtract $6 x^{3}$ from $6 x^{3}$ .

$4 x^{2} - 6 x - 4 + 0 - 4 x^{2} + 6 k^{2} x = - 4 k^{2}$

Step 2.4.2

Add $4 x^{2} - 6 x - 4$ and $0$ .

$4 x^{2} - 6 x - 4 - 4 x^{2} + 6 k^{2} x = - 4 k^{2}$

Step 2.4.3

Subtract $4 x^{2}$ from $4 x^{2}$ .

$- 6 x - 4 + 0 + 6 k^{2} x = - 4 k^{2}$

Step 2.4.4

Add $- 6 x - 4$ and $0$ .

$- 6 x - 4 + 6 k^{2} x = - 4 k^{2}$

Step 3

Add $4$ to both sides of the equation.

$- 6 x + 6 k^{2} x = - 4 k^{2} + 4$

Step 4

Factor $6 x$ out of $- 6 x + 6 k^{2} x$ .

Tap for more steps...

Step 4.1

Factor $6 x$ out of $- 6 x$ .

$6 x (- 1) + 6 k^{2} x = - 4 k^{2} + 4$

Step 4.2

Factor $6 x$ out of $6 k^{2} x$ .

$6 x (- 1) + 6 x (k^{2}) = - 4 k^{2} + 4$

Step 4.3

Factor $6 x$ out of $6 x (- 1) + 6 x (k^{2})$ .

$6 x (- 1 + k^{2}) = - 4 k^{2} + 4$

Step 5

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

$6 x (- 1^{2} + k^{2}) = - 4 k^{2} + 4$

Step 6

Reorder $- 1^{2}$ and $k^{2}$ .

$6 x (k^{2} - 1^{2}) = - 4 k^{2} + 4$

Step 7

Factor.

Tap for more steps...

Step 7.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = k$ and $b = 1$ .

$6 x ((k + 1) (k - 1)) = - 4 k^{2} + 4$

Step 7.2

Remove unnecessary parentheses.

$6 x (k + 1) (k - 1) = - 4 k^{2} + 4$

Step 8

Divide each term in $6 x (k + 1) (k - 1) = - 4 k^{2} + 4$ by $6 (k + 1) (k - 1)$ and simplify.

Tap for more steps...

Step 8.1

Divide each term in $6 x (k + 1) (k - 1) = - 4 k^{2} + 4$ by $6 (k + 1) (k - 1)$ .

$\frac{6 x (k + 1) (k - 1)}{6 (k + 1) (k - 1)} = \frac{- 4 k^{2}}{6 (k + 1) (k - 1)} + \frac{4}{6 (k + 1) (k - 1)}$

Step 8.2

Simplify the left side.

Tap for more steps...

Step 8.2.1

Cancel the common factor of $6$ .

Tap for more steps...

Step 8.2.1.1

Cancel the common factor.

$\frac{6 x (k + 1) (k - 1)}{6 (k + 1) (k - 1)} = \frac{- 4 k^{2}}{6 (k + 1) (k - 1)} + \frac{4}{6 (k + 1) (k - 1)}$

Step 8.2.1.2

Rewrite the expression.

$\frac{(x (k + 1)) (k - 1)}{(k + 1) (k - 1)} = \frac{- 4 k^{2}}{6 (k + 1) (k - 1)} + \frac{4}{6 (k + 1) (k - 1)}$

Step 8.2.2

Cancel the common factor of $k + 1$ .

Tap for more steps...

Step 8.2.2.1

Cancel the common factor.

$\frac{x (k + 1) (k - 1)}{(k + 1) (k - 1)} = \frac{- 4 k^{2}}{6 (k + 1) (k - 1)} + \frac{4}{6 (k + 1) (k - 1)}$

Step 8.2.2.2

Rewrite the expression.

$\frac{(x) (k - 1)}{k - 1} = \frac{- 4 k^{2}}{6 (k + 1) (k - 1)} + \frac{4}{6 (k + 1) (k - 1)}$

Step 8.2.3

Cancel the common factor of $k - 1$ .

Tap for more steps...

Step 8.2.3.1

Cancel the common factor.

$\frac{x (k - 1)}{k - 1} = \frac{- 4 k^{2}}{6 (k + 1) (k - 1)} + \frac{4}{6 (k + 1) (k - 1)}$

Step 8.2.3.2

Divide $x$ by $1$ .

$x = \frac{- 4 k^{2}}{6 (k + 1) (k - 1)} + \frac{4}{6 (k + 1) (k - 1)}$

Step 8.3

Simplify the right side.

Tap for more steps...

Step 8.3.1

Simplify each term.

Tap for more steps...

Step 8.3.1.1

Cancel the common factor of $- 4$ and $6$ .

Tap for more steps...

Step 8.3.1.1.1

Factor $2$ out of $- 4 k^{2}$ .

$x = \frac{2 (- 2 k^{2})}{6 (k + 1) (k - 1)} + \frac{4}{6 (k + 1) (k - 1)}$

Step 8.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 8.3.1.1.2.1

Factor $2$ out of $6 (k + 1) (k - 1)$ .

$x = \frac{2 (- 2 k^{2})}{2 (3 (k + 1) (k - 1))} + \frac{4}{6 (k + 1) (k - 1)}$

Step 8.3.1.1.2.2

Cancel the common factor.

$x = \frac{2 (- 2 k^{2})}{2 (3 (k + 1) (k - 1))} + \frac{4}{6 (k + 1) (k - 1)}$

Step 8.3.1.1.2.3

Rewrite the expression.

$x = \frac{- 2 k^{2}}{3 (k + 1) (k - 1)} + \frac{4}{6 (k + 1) (k - 1)}$

Step 8.3.1.2

Move the negative in front of the fraction.

$x = - \frac{(2) k^{2}}{3 (k + 1) (k - 1)} + \frac{4}{6 (k + 1) (k - 1)}$

Step 8.3.1.3

Cancel the common factor of $4$ and $6$ .

Tap for more steps...

Step 8.3.1.3.1

Factor $2$ out of $4$ .

$x = - \frac{2 k^{2}}{3 (k + 1) (k - 1)} + \frac{2 (2)}{6 (k + 1) (k - 1)}$

Step 8.3.1.3.2

Cancel the common factors.

Tap for more steps...

Step 8.3.1.3.2.1

Factor $2$ out of $6 (k + 1) (k - 1)$ .

$x = - \frac{2 k^{2}}{3 (k + 1) (k - 1)} + \frac{2 (2)}{2 (3 (k + 1) (k - 1))}$

Step 8.3.1.3.2.2

Cancel the common factor.

$x = - \frac{2 k^{2}}{3 (k + 1) (k - 1)} + \frac{2 \cdot 2}{2 (3 (k + 1) (k - 1))}$

Step 8.3.1.3.2.3

Rewrite the expression.

$x = - \frac{2 k^{2}}{3 (k + 1) (k - 1)} + \frac{2}{3 (k + 1) (k - 1)}$

Step 9

To rewrite as a function of $k$ , write the equation so that $x$ is by itself on one side of the equal sign and an expression involving only $k$ is on the other side.

$f (k) = - \frac{2 k^{2}}{3 (k + 1) (k - 1)} + \frac{2}{3 (k + 1) (k - 1)}$