Algebra Examples

$3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4) = 5 x (x^{2} - 2 x + 3) - x (2 x^{2} + x) - 1$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$5 x (x^{2} - 2 x + 3) - x (2 x^{2} + x) - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 2

Simplify $5 x (x^{2} - 2 x + 3) - x (2 x^{2} + x) - 1$ .

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Apply the distributive property.

$5 x \cdot x^{2} + 5 x (- 2 x) + 5 x \cdot 3 - x (2 x^{2} + x) - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 2.1.2

Simplify.

Tap for more steps...

Step 2.1.2.1

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

Tap for more steps...

Step 2.1.2.1.1

Move $x^{2}$ .

$5 (x^{2} x) + 5 x (- 2 x) + 5 x \cdot 3 - x (2 x^{2} + x) - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 2.1.2.1.2

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

Tap for more steps...

Step 2.1.2.1.2.1

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

$5 (x^{2} x^{1}) + 5 x (- 2 x) + 5 x \cdot 3 - x (2 x^{2} + x) - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 2.1.2.1.2.2

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

$5 x^{2 + 1} + 5 x (- 2 x) + 5 x \cdot 3 - x (2 x^{2} + x) - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 2.1.2.1.3

Add $2$ and $1$ .

$5 x^{3} + 5 x (- 2 x) + 5 x \cdot 3 - x (2 x^{2} + x) - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 2.1.2.2

Rewrite using the commutative property of multiplication.

$5 x^{3} + 5 \cdot - 2 x \cdot x + 5 x \cdot 3 - x (2 x^{2} + x) - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 2.1.2.3

Multiply $3$ by $5$ .

$5 x^{3} + 5 \cdot - 2 x \cdot x + 15 x - x (2 x^{2} + x) - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 2.1.3

Simplify each term.

Tap for more steps...

Step 2.1.3.1

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

Tap for more steps...

Step 2.1.3.1.1

Move $x$ .

$5 x^{3} + 5 \cdot - 2 (x \cdot x) + 15 x - x (2 x^{2} + x) - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 2.1.3.1.2

Multiply $x$ by $x$ .

$5 x^{3} + 5 \cdot - 2 x^{2} + 15 x - x (2 x^{2} + x) - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 2.1.3.2

Multiply $5$ by $- 2$ .

$5 x^{3} - 10 x^{2} + 15 x - x (2 x^{2} + x) - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 2.1.4

Apply the distributive property.

$5 x^{3} - 10 x^{2} + 15 x - x (2 x^{2}) - x \cdot x - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 2.1.5

Rewrite using the commutative property of multiplication.

$5 x^{3} - 10 x^{2} + 15 x - 1 \cdot 2 x \cdot x^{2} - x \cdot x - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 2.1.6

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

Tap for more steps...

Step 2.1.6.1

Move $x$ .

$5 x^{3} - 10 x^{2} + 15 x - 1 \cdot 2 x \cdot x^{2} - (x \cdot x) - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 2.1.6.2

Multiply $x$ by $x$ .

$5 x^{3} - 10 x^{2} + 15 x - 1 \cdot 2 x \cdot x^{2} - x^{2} - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 2.1.7

Simplify each term.

Tap for more steps...

Step 2.1.7.1

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

Tap for more steps...

Step 2.1.7.1.1

Move $x^{2}$ .

$5 x^{3} - 10 x^{2} + 15 x - 1 \cdot 2 (x^{2} x) - x^{2} - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 2.1.7.1.2

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

Tap for more steps...

Step 2.1.7.1.2.1

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

$5 x^{3} - 10 x^{2} + 15 x - 1 \cdot 2 (x^{2} x^{1}) - x^{2} - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 2.1.7.1.2.2

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

$5 x^{3} - 10 x^{2} + 15 x - 1 \cdot 2 x^{2 + 1} - x^{2} - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 2.1.7.1.3

Add $2$ and $1$ .

$5 x^{3} - 10 x^{2} + 15 x - 1 \cdot 2 x^{3} - x^{2} - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 2.1.7.2

Multiply $- 1$ by $2$ .

$5 x^{3} - 10 x^{2} + 15 x - 2 x^{3} - x^{2} - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 2.2

Simplify by adding terms.

Tap for more steps...

Step 2.2.1

Subtract $2 x^{3}$ from $5 x^{3}$ .

$3 x^{3} - 10 x^{2} + 15 x - x^{2} - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 2.2.2

Subtract $x^{2}$ from $- 10 x^{2}$ .

$3 x^{3} - 11 x^{2} + 15 x - 1 = 3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$

Step 3

Simplify $3 x (x^{2} - 5 x + 6) + 4 (x^{2} + 3 x - 4)$ .

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

Apply the distributive property.

$3 x^{3} - 11 x^{2} + 15 x - 1 = 3 x \cdot x^{2} + 3 x (- 5 x) + 3 x \cdot 6 + 4 (x^{2} + 3 x - 4)$

Step 3.1.2

Simplify.

Tap for more steps...

Step 3.1.2.1

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

Tap for more steps...

Step 3.1.2.1.1

Move $x^{2}$ .

$3 x^{3} - 11 x^{2} + 15 x - 1 = 3 (x^{2} x) + 3 x (- 5 x) + 3 x \cdot 6 + 4 (x^{2} + 3 x - 4)$

Step 3.1.2.1.2

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

Tap for more steps...

Step 3.1.2.1.2.1

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

$3 x^{3} - 11 x^{2} + 15 x - 1 = 3 (x^{2} x^{1}) + 3 x (- 5 x) + 3 x \cdot 6 + 4 (x^{2} + 3 x - 4)$

Step 3.1.2.1.2.2

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

$3 x^{3} - 11 x^{2} + 15 x - 1 = 3 x^{2 + 1} + 3 x (- 5 x) + 3 x \cdot 6 + 4 (x^{2} + 3 x - 4)$

Step 3.1.2.1.3

Add $2$ and $1$ .

$3 x^{3} - 11 x^{2} + 15 x - 1 = 3 x^{3} + 3 x (- 5 x) + 3 x \cdot 6 + 4 (x^{2} + 3 x - 4)$

Step 3.1.2.2

Rewrite using the commutative property of multiplication.

$3 x^{3} - 11 x^{2} + 15 x - 1 = 3 x^{3} + 3 \cdot - 5 x \cdot x + 3 x \cdot 6 + 4 (x^{2} + 3 x - 4)$

Step 3.1.2.3

Multiply $6$ by $3$ .

$3 x^{3} - 11 x^{2} + 15 x - 1 = 3 x^{3} + 3 \cdot - 5 x \cdot x + 18 x + 4 (x^{2} + 3 x - 4)$

Step 3.1.3

Simplify each term.

Tap for more steps...

Step 3.1.3.1

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

Tap for more steps...

Step 3.1.3.1.1

Move $x$ .

$3 x^{3} - 11 x^{2} + 15 x - 1 = 3 x^{3} + 3 \cdot - 5 (x \cdot x) + 18 x + 4 (x^{2} + 3 x - 4)$

Step 3.1.3.1.2

Multiply $x$ by $x$ .

$3 x^{3} - 11 x^{2} + 15 x - 1 = 3 x^{3} + 3 \cdot - 5 x^{2} + 18 x + 4 (x^{2} + 3 x - 4)$

Step 3.1.3.2

Multiply $3$ by $- 5$ .

$3 x^{3} - 11 x^{2} + 15 x - 1 = 3 x^{3} - 15 x^{2} + 18 x + 4 (x^{2} + 3 x - 4)$

Step 3.1.4

Apply the distributive property.

$3 x^{3} - 11 x^{2} + 15 x - 1 = 3 x^{3} - 15 x^{2} + 18 x + 4 x^{2} + 4 (3 x) + 4 \cdot - 4$

Step 3.1.5

Simplify.

Tap for more steps...

Step 3.1.5.1

Multiply $3$ by $4$ .

$3 x^{3} - 11 x^{2} + 15 x - 1 = 3 x^{3} - 15 x^{2} + 18 x + 4 x^{2} + 12 x + 4 \cdot - 4$

Step 3.1.5.2

Multiply $4$ by $- 4$ .

$3 x^{3} - 11 x^{2} + 15 x - 1 = 3 x^{3} - 15 x^{2} + 18 x + 4 x^{2} + 12 x - 16$

Step 3.2

Simplify by adding terms.

Tap for more steps...

Step 3.2.1

Add $- 15 x^{2}$ and $4 x^{2}$ .

$3 x^{3} - 11 x^{2} + 15 x - 1 = 3 x^{3} - 11 x^{2} + 18 x + 12 x - 16$

Step 3.2.2

Add $18 x$ and $12 x$ .

$3 x^{3} - 11 x^{2} + 15 x - 1 = 3 x^{3} - 11 x^{2} + 30 x - 16$

Step 4

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

Tap for more steps...

Step 4.1

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

$3 x^{3} - 11 x^{2} + 15 x - 1 - 3 x^{3} = - 11 x^{2} + 30 x - 16$

Step 4.2

Add $11 x^{2}$ to both sides of the equation.

$3 x^{3} - 11 x^{2} + 15 x - 1 - 3 x^{3} + 11 x^{2} = 30 x - 16$

Step 4.3

Subtract $30 x$ from both sides of the equation.

$3 x^{3} - 11 x^{2} + 15 x - 1 - 3 x^{3} + 11 x^{2} - 30 x = - 16$

Step 4.4

Combine the opposite terms in $3 x^{3} - 11 x^{2} + 15 x - 1 - 3 x^{3} + 11 x^{2} - 30 x$ .

Tap for more steps...

Step 4.4.1

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

$- 11 x^{2} + 15 x - 1 + 0 + 11 x^{2} - 30 x = - 16$

Step 4.4.2

Add $- 11 x^{2} + 15 x - 1$ and $0$ .

$- 11 x^{2} + 15 x - 1 + 11 x^{2} - 30 x = - 16$

Step 4.4.3

Add $- 11 x^{2}$ and $11 x^{2}$ .

$15 x - 1 + 0 - 30 x = - 16$

Step 4.4.4

Add $15 x - 1$ and $0$ .

$15 x - 1 - 30 x = - 16$

Step 4.5

Subtract $30 x$ from $15 x$ .

$- 15 x - 1 = - 16$

Step 5

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 5.1

Add $1$ to both sides of the equation.

$- 15 x = - 16 + 1$

Step 5.2

Add $- 16$ and $1$ .

$- 15 x = - 15$

Step 6

Divide each term in $- 15 x = - 15$ by $- 15$ and simplify.

Tap for more steps...

Step 6.1

Divide each term in $- 15 x = - 15$ by $- 15$ .

$\frac{- 15 x}{- 15} = \frac{- 15}{- 15}$

Step 6.2

Simplify the left side.

Tap for more steps...

Step 6.2.1

Cancel the common factor of $- 15$ .

Tap for more steps...

Step 6.2.1.1

Cancel the common factor.

$\frac{- 15 x}{- 15} = \frac{- 15}{- 15}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 15}{- 15}$

Step 6.3

Simplify the right side.

Tap for more steps...

Step 6.3.1

Divide $- 15$ by $- 15$ .

$x = 1$