Algebra Examples

$2 x^{3} - 2 x^{2} - 24 x = a x (x + b) (x + c)$

Step 1

Simplify $a x (x + b) (x + c)$ .

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Step 1.1

Apply the distributive property.

$2 x^{3} - 2 x^{2} - 24 x = (a x \cdot x + a x b) (x + c)$

Step 1.2

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

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Step 1.2.1

Move $x$ .

$2 x^{3} - 2 x^{2} - 24 x = (a (x \cdot x) + a x b) (x + c)$

Step 1.2.2

Multiply $x$ by $x$ .

$2 x^{3} - 2 x^{2} - 24 x = (a x^{2} + a x b) (x + c)$

Step 1.3

Expand $(a x^{2} + a x b) (x + c)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$2 x^{3} - 2 x^{2} - 24 x = a x^{2} (x + c) + a x b (x + c)$

Step 1.3.2

Apply the distributive property.

$2 x^{3} - 2 x^{2} - 24 x = a x^{2} x + a x^{2} c + a x b (x + c)$

Step 1.3.3

Apply the distributive property.

$2 x^{3} - 2 x^{2} - 24 x = a x^{2} x + a x^{2} c + a x b x + a x b c$

Step 1.4

Simplify each term.

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Step 1.4.1

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

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Step 1.4.1.1

Move $x$ .

$2 x^{3} - 2 x^{2} - 24 x = a (x \cdot x^{2}) + a x^{2} c + a x b x + a x b c$

Step 1.4.1.2

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

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Step 1.4.1.2.1

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

$2 x^{3} - 2 x^{2} - 24 x = a (x^{1} x^{2}) + a x^{2} c + a x b x + a x b c$

Step 1.4.1.2.2

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

$2 x^{3} - 2 x^{2} - 24 x = a x^{1 + 2} + a x^{2} c + a x b x + a x b c$

Step 1.4.1.3

Add $1$ and $2$ .

$2 x^{3} - 2 x^{2} - 24 x = a x^{3} + a x^{2} c + a x b x + a x b c$

Step 1.4.2

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

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Step 1.4.2.1

Move $x$ .

$2 x^{3} - 2 x^{2} - 24 x = a x^{3} + a x^{2} c + a (x \cdot x) b + a x b c$

Step 1.4.2.2

Multiply $x$ by $x$ .

$2 x^{3} - 2 x^{2} - 24 x = a x^{3} + a x^{2} c + a x^{2} b + a x b c$

Step 2

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

$a x^{3} + a x^{2} c + a x^{2} b + a x b c = 2 x^{3} - 2 x^{2} - 24 x$

Step 3

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

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Step 3.1

Subtract $2 x^{3}$ from both sides of the equation.

$a x^{3} + a x^{2} c + a x^{2} b + a x b c - 2 x^{3} = - 2 x^{2} - 24 x$

Step 3.2

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

$a x^{3} + a x^{2} c + a x^{2} b + a x b c - 2 x^{3} + 2 x^{2} = - 24 x$

Step 3.3

Add $24 x$ to both sides of the equation.

$a x^{3} + a x^{2} c + a x^{2} b + a x b c - 2 x^{3} + 2 x^{2} + 24 x = 0$

Step 4

Factor $x$ out of $a x^{3} + a x^{2} c + a x^{2} b + a x b c - 2 x^{3} + 2 x^{2} + 24 x$ .

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Step 4.1

Factor $x$ out of $a x^{3}$ .

$x (a x^{2}) + a x^{2} c + a x^{2} b + a x b c - 2 x^{3} + 2 x^{2} + 24 x = 0$

Step 4.2

Factor $x$ out of $a x^{2} c$ .

$x (a x^{2}) + x (a x c) + a x^{2} b + a x b c - 2 x^{3} + 2 x^{2} + 24 x = 0$

Step 4.3

Factor $x$ out of $a x^{2} b$ .

$x (a x^{2}) + x (a x c) + x (a x b) + a x b c - 2 x^{3} + 2 x^{2} + 24 x = 0$

Step 4.4

Factor $x$ out of $a x b c$ .

$x (a x^{2}) + x (a x c) + x (a x b) + x (a b c) - 2 x^{3} + 2 x^{2} + 24 x = 0$

Step 4.5

Factor $x$ out of $- 2 x^{3}$ .

$x (a x^{2}) + x (a x c) + x (a x b) + x (a b c) + x (- 2 x^{2}) + 2 x^{2} + 24 x = 0$

Step 4.6

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

$x (a x^{2}) + x (a x c) + x (a x b) + x (a b c) + x (- 2 x^{2}) + x (2 x) + 24 x = 0$

Step 4.7

Factor $x$ out of $24 x$ .

$x (a x^{2}) + x (a x c) + x (a x b) + x (a b c) + x (- 2 x^{2}) + x (2 x) + x \cdot 24 = 0$

Step 4.8

Factor $x$ out of $x (a x^{2}) + x (a x c)$ .

$x (a x^{2} + a x c) + x (a x b) + x (a b c) + x (- 2 x^{2}) + x (2 x) + x \cdot 24 = 0$

Step 4.9

Factor $x$ out of $x (a x^{2} + a x c) + x (a x b)$ .

$x (a x^{2} + a x c + a x b) + x (a b c) + x (- 2 x^{2}) + x (2 x) + x \cdot 24 = 0$

Step 4.10

Factor $x$ out of $x (a x^{2} + a x c + a x b) + x (a b c)$ .

$x (a x^{2} + a x c + a x b + a b c) + x (- 2 x^{2}) + x (2 x) + x \cdot 24 = 0$

Step 4.11

Factor $x$ out of $x (a x^{2} + a x c + a x b + a b c) + x (- 2 x^{2})$ .

$x (a x^{2} + a x c + a x b + a b c - 2 x^{2}) + x (2 x) + x \cdot 24 = 0$

Step 4.12

Factor $x$ out of $x (a x^{2} + a x c + a x b + a b c - 2 x^{2}) + x (2 x)$ .

$x (a x^{2} + a x c + a x b + a b c - 2 x^{2} + 2 x) + x \cdot 24 = 0$

Step 4.13

Factor $x$ out of $x (a x^{2} + a x c + a x b + a b c - 2 x^{2} + 2 x) + x \cdot 24$ .

$x (a x^{2} + a x c + a x b + a b c - 2 x^{2} + 2 x + 24) = 0$

Step 5

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

$x = 0$

$a x^{2} + a x c + a x b + a b c - 2 x^{2} + 2 x + 24 = 0$

Step 6

Set $x$ equal to $0$ .

$x = 0$

Step 7

Set $a x^{2} + a x c + a x b + a b c - 2 x^{2} + 2 x + 24$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $a x^{2} + a x c + a x b + a b c - 2 x^{2} + 2 x + 24$ equal to $0$ .

$a x^{2} + a x c + a x b + a b c - 2 x^{2} + 2 x + 24 = 0$

Step 7.2

Solve $a x^{2} + a x c + a x b + a b c - 2 x^{2} + 2 x + 24 = 0$ for $x$ .

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Step 7.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7.2.2

Substitute the values $a = a - 2$ , $b = a c + a b + 2$ , and $c = a b c + 24$ into the quadratic formula and solve for $x$ .

$\frac{- (a c + a b + 2) \pm \sqrt{{(a c + a b + 2)}^{2} - 4 \cdot ((a - 2) \cdot (a b c + 24))}}{2 (a - 2)}$

Step 7.2.3

Simplify the numerator.

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Step 7.2.3.1

Apply the distributive property.

$x = \frac{- (a c) - (a b) - 1 \cdot 2 \pm \sqrt{{(a c + a b + 2)}^{2} - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.2

Multiply $- 1$ by $2$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{{(a c + a b + 2)}^{2} - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.3

Rewrite ${(a c + a b + 2)}^{2}$ as $(a c + a b + 2) (a c + a b + 2)$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{(a c + a b + 2) (a c + a b + 2) - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.4

Expand $(a c + a b + 2) (a c + a b + 2)$ by multiplying each term in the first expression by each term in the second expression.

$x = \frac{- a c - a b - 2 \pm \sqrt{a c (a c) + a c (a b) + a c \cdot 2 + a b (a c) + a b (a b) + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.5

Simplify each term.

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Step 7.2.3.5.1

Multiply $a$ by $a$ by adding the exponents.

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Step 7.2.3.5.1.1

Move $a$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a \cdot a c \cdot c + a c (a b) + a c \cdot 2 + a b (a c) + a b (a b) + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.5.1.2

Multiply $a$ by $a$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c \cdot c + a c (a b) + a c \cdot 2 + a b (a c) + a b (a b) + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.5.2

Multiply $c$ by $c$ by adding the exponents.

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Step 7.2.3.5.2.1

Move $c$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} (c \cdot c) + a c (a b) + a c \cdot 2 + a b (a c) + a b (a b) + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.5.2.2

Multiply $c$ by $c$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a c (a b) + a c \cdot 2 + a b (a c) + a b (a b) + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.5.3

Multiply $a$ by $a$ by adding the exponents.

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Step 7.2.3.5.3.1

Move $a$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a \cdot a c b + a c \cdot 2 + a b (a c) + a b (a b) + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.5.3.2

Multiply $a$ by $a$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} c b + a c \cdot 2 + a b (a c) + a b (a b) + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.5.4

Move $2$ to the left of $a c$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} c b + 2 \cdot (a c) + a b (a c) + a b (a b) + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.5.5

Multiply $a$ by $a$ by adding the exponents.

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Step 7.2.3.5.5.1

Move $a$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} c b + 2 a c + a \cdot a b c + a b (a b) + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.5.5.2

Multiply $a$ by $a$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} c b + 2 a c + a^{2} b c + a b (a b) + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.5.6

Multiply $a$ by $a$ by adding the exponents.

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Step 7.2.3.5.6.1

Move $a$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} c b + 2 a c + a^{2} b c + a \cdot a b \cdot b + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.5.6.2

Multiply $a$ by $a$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} c b + 2 a c + a^{2} b c + a^{2} b \cdot b + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.5.7

Multiply $b$ by $b$ by adding the exponents.

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Step 7.2.3.5.7.1

Move $b$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} c b + 2 a c + a^{2} b c + a^{2} (b \cdot b) + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.5.7.2

Multiply $b$ by $b$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} c b + 2 a c + a^{2} b c + a^{2} b^{2} + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.5.8

Move $2$ to the left of $a b$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} c b + 2 a c + a^{2} b c + a^{2} b^{2} + 2 \cdot (a b) + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.5.9

Multiply $2$ by $2$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} c b + 2 a c + a^{2} b c + a^{2} b^{2} + 2 a b + 2 a c + 2 a b + 4 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.6

Add $a^{2} c b$ and $a^{2} b c$ .

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Step 7.2.3.6.1

Move $c$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} b c + a^{2} b c + 2 a c + a^{2} b^{2} + 2 a b + 2 a c + 2 a b + 4 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.6.2

Add $a^{2} b c$ and $a^{2} b c$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 2 a c + a^{2} b^{2} + 2 a b + 2 a c + 2 a b + 4 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.7

Add $2 a c$ and $2 a c$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 2 a b + 2 a b + 4 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.8

Add $2 a b$ and $2 a b$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.9

Apply the distributive property.

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 + (- 4 a - 4 \cdot - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.10

Multiply $- 4$ by $- 2$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 + (- 4 a + 8) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.11

Expand $(- 4 a + 8) (a b c + 24)$ using the FOIL Method.

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Step 7.2.3.11.1

Apply the distributive property.

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 - 4 a (a b c + 24) + 8 (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.11.2

Apply the distributive property.

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 - 4 a (a b c) - 4 a \cdot 24 + 8 (a b c + 24)}}{2 (a - 2)}$

Step 7.2.3.11.3

Apply the distributive property.

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 - 4 a (a b c) - 4 a \cdot 24 + 8 (a b c) + 8 \cdot 24}}{2 (a - 2)}$

Step 7.2.3.12

Simplify each term.

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Step 7.2.3.12.1

Multiply $a$ by $a$ by adding the exponents.

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Step 7.2.3.12.1.1

Move $a$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 - 4 (a \cdot a) (b c) - 4 a \cdot 24 + 8 (a b c) + 8 \cdot 24}}{2 (a - 2)}$

Step 7.2.3.12.1.2

Multiply $a$ by $a$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 - 4 a^{2} (b c) - 4 a \cdot 24 + 8 (a b c) + 8 \cdot 24}}{2 (a - 2)}$

Step 7.2.3.12.2

Multiply $24$ by $- 4$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 - 4 a^{2} b c - 96 a + 8 (a b c) + 8 \cdot 24}}{2 (a - 2)}$

Step 7.2.3.12.3

Multiply $8$ by $24$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 - 4 a^{2} b c - 96 a + 8 a b c + 192}}{2 (a - 2)}$

Step 7.2.3.13

Subtract $4 a^{2} b c$ from $2 a^{2} b c$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 - 96 a + 8 a b c + 192}}{2 (a - 2)}$

Step 7.2.3.14

Add $4$ and $192$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

Step 7.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 7.2.4.1

Change the $\pm$ to $+$ .

$x = \frac{- a c - a b - 2 + \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

Step 7.2.4.2

Factor $- 1$ out of $- a c$ .

$x = \frac{- (a c) - a b - 2 + \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

Step 7.2.4.3

Factor $- 1$ out of $- a b$ .

$x = \frac{- (a c) - (a b) - 2 + \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

Step 7.2.4.4

Factor $- 1$ out of $- (a c) - (a b)$ .

$x = \frac{- (a c + a b) - 2 + \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

Step 7.2.4.5

Rewrite $- 2$ as $- 1 (2)$ .

$x = \frac{- (a c + a b) - 1 \cdot 2 + \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

Step 7.2.4.6

Factor $- 1$ out of $- (a c + a b) - 1 (2)$ .

$x = \frac{- (a c + a b + 2) + \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

Step 7.2.4.7

Factor $- 1$ out of $\sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}$ .

$x = \frac{- (a c + a b + 2) - 1 (- \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196})}{2 (a - 2)}$

Step 7.2.4.8

Factor $- 1$ out of $- (a c + a b + 2) - 1 (- \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196})$ .

$x = \frac{- (a c + a b + 2 - \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196})}{2 (a - 2)}$

Step 7.2.4.9

Rewrite $- (a c + a b + 2 - \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196})$ as $- 1 (a c + a b + 2 - \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196})$ .

$x = \frac{- 1 (a c + a b + 2 - \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196})}{2 (a - 2)}$

Step 7.2.4.10

Move the negative in front of the fraction.

$x = - \frac{a c + a b + 2 - \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

Step 7.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.2.5.1

Simplify the numerator.

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Step 7.2.5.1.1

Apply the distributive property.

$x = \frac{- (a c) - (a b) - 1 \cdot 2 \pm \sqrt{{(a c + a b + 2)}^{2} - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.2

Multiply $- 1$ by $2$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{{(a c + a b + 2)}^{2} - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.3

Rewrite ${(a c + a b + 2)}^{2}$ as $(a c + a b + 2) (a c + a b + 2)$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{(a c + a b + 2) (a c + a b + 2) - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.4

Expand $(a c + a b + 2) (a c + a b + 2)$ by multiplying each term in the first expression by each term in the second expression.

$x = \frac{- a c - a b - 2 \pm \sqrt{a c (a c) + a c (a b) + a c \cdot 2 + a b (a c) + a b (a b) + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.5

Simplify each term.

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Step 7.2.5.1.5.1

Multiply $a$ by $a$ by adding the exponents.

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Step 7.2.5.1.5.1.1

Move $a$ .

Step 7.2.5.1.5.1.2

Multiply $a$ by $a$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c \cdot c + a c (a b) + a c \cdot 2 + a b (a c) + a b (a b) + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.5.2

Multiply $c$ by $c$ by adding the exponents.

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Step 7.2.5.1.5.2.1

Move $c$ .

Step 7.2.5.1.5.2.2

Multiply $c$ by $c$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a c (a b) + a c \cdot 2 + a b (a c) + a b (a b) + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.5.3

Multiply $a$ by $a$ by adding the exponents.

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Step 7.2.5.1.5.3.1

Move $a$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a \cdot a c b + a c \cdot 2 + a b (a c) + a b (a b) + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.5.3.2

Multiply $a$ by $a$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} c b + a c \cdot 2 + a b (a c) + a b (a b) + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.5.4

Move $2$ to the left of $a c$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} c b + 2 \cdot (a c) + a b (a c) + a b (a b) + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.5.5

Multiply $a$ by $a$ by adding the exponents.

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Step 7.2.5.1.5.5.1

Move $a$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} c b + 2 a c + a \cdot a b c + a b (a b) + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.5.5.2

Multiply $a$ by $a$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} c b + 2 a c + a^{2} b c + a b (a b) + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.5.6

Multiply $a$ by $a$ by adding the exponents.

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Step 7.2.5.1.5.6.1

Move $a$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} c b + 2 a c + a^{2} b c + a \cdot a b \cdot b + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.5.6.2

Multiply $a$ by $a$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} c b + 2 a c + a^{2} b c + a^{2} b \cdot b + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.5.7

Multiply $b$ by $b$ by adding the exponents.

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Step 7.2.5.1.5.7.1

Move $b$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} c b + 2 a c + a^{2} b c + a^{2} (b \cdot b) + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.5.7.2

Multiply $b$ by $b$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} c b + 2 a c + a^{2} b c + a^{2} b^{2} + a b \cdot 2 + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.5.8

Move $2$ to the left of $a b$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} c b + 2 a c + a^{2} b c + a^{2} b^{2} + 2 \cdot (a b) + 2 (a c) + 2 (a b) + 2 \cdot 2 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.5.9

Multiply $2$ by $2$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} c b + 2 a c + a^{2} b c + a^{2} b^{2} + 2 a b + 2 a c + 2 a b + 4 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.6

Add $a^{2} c b$ and $a^{2} b c$ .

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Step 7.2.5.1.6.1

Move $c$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + a^{2} b c + a^{2} b c + 2 a c + a^{2} b^{2} + 2 a b + 2 a c + 2 a b + 4 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.6.2

Add $a^{2} b c$ and $a^{2} b c$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 2 a c + a^{2} b^{2} + 2 a b + 2 a c + 2 a b + 4 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.7

Add $2 a c$ and $2 a c$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 2 a b + 2 a b + 4 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.8

Add $2 a b$ and $2 a b$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 - 4 \cdot (a - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.9

Apply the distributive property.

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 + (- 4 a - 4 \cdot - 2) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.10

Multiply $- 4$ by $- 2$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 + (- 4 a + 8) \cdot (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.11

Expand $(- 4 a + 8) (a b c + 24)$ using the FOIL Method.

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Step 7.2.5.1.11.1

Apply the distributive property.

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 - 4 a (a b c + 24) + 8 (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.11.2

Apply the distributive property.

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 - 4 a (a b c) - 4 a \cdot 24 + 8 (a b c + 24)}}{2 (a - 2)}$

Step 7.2.5.1.11.3

Apply the distributive property.

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 - 4 a (a b c) - 4 a \cdot 24 + 8 (a b c) + 8 \cdot 24}}{2 (a - 2)}$

Step 7.2.5.1.12

Simplify each term.

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Step 7.2.5.1.12.1

Multiply $a$ by $a$ by adding the exponents.

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Step 7.2.5.1.12.1.1

Move $a$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 - 4 (a \cdot a) (b c) - 4 a \cdot 24 + 8 (a b c) + 8 \cdot 24}}{2 (a - 2)}$

Step 7.2.5.1.12.1.2

Multiply $a$ by $a$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 - 4 a^{2} (b c) - 4 a \cdot 24 + 8 (a b c) + 8 \cdot 24}}{2 (a - 2)}$

Step 7.2.5.1.12.2

Multiply $24$ by $- 4$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 - 4 a^{2} b c - 96 a + 8 (a b c) + 8 \cdot 24}}{2 (a - 2)}$

Step 7.2.5.1.12.3

Multiply $8$ by $24$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} + 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 - 4 a^{2} b c - 96 a + 8 a b c + 192}}{2 (a - 2)}$

Step 7.2.5.1.13

Subtract $4 a^{2} b c$ from $2 a^{2} b c$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b + 4 - 96 a + 8 a b c + 192}}{2 (a - 2)}$

Step 7.2.5.1.14

Add $4$ and $192$ .

$x = \frac{- a c - a b - 2 \pm \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

Step 7.2.5.2

Change the $\pm$ to $-$ .

$x = \frac{- a c - a b - 2 - \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

Step 7.2.5.3

Factor $- 1$ out of $- a c$ .

$x = \frac{- (a c) - a b - 2 - \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

Step 7.2.5.4

Factor $- 1$ out of $- a b$ .

$x = \frac{- (a c) - (a b) - 2 - \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

Step 7.2.5.5

Factor $- 1$ out of $- (a c) - (a b)$ .

$x = \frac{- (a c + a b) - 2 - \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

Step 7.2.5.6

Rewrite $- 2$ as $- 1 (2)$ .

$x = \frac{- (a c + a b) - 1 \cdot 2 - \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

Step 7.2.5.7

Factor $- 1$ out of $- (a c + a b) - 1 (2)$ .

$x = \frac{- (a c + a b + 2) - \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

Step 7.2.5.8

Factor $- 1$ out of $- \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}$ .

$x = \frac{- (a c + a b + 2) - (\sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196})}{2 (a - 2)}$

Step 7.2.5.9

Factor $- 1$ out of $- (a c + a b + 2) - (\sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196})$ .

$x = \frac{- (a c + a b + 2 + \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196})}{2 (a - 2)}$

Step 7.2.5.10

Rewrite $- (a c + a b + 2 + \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196})$ as $- 1 (a c + a b + 2 + \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196})$ .

$x = \frac{- 1 (a c + a b + 2 + \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196})}{2 (a - 2)}$

Step 7.2.5.11

Move the negative in front of the fraction.

$x = - \frac{a c + a b + 2 + \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

Step 7.2.6

The final answer is the combination of both solutions.

$x = - \frac{a c + a b + 2 - \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

$x = - \frac{a c + a b + 2 + \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

$x = - \frac{a c + a b + 2 - \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

$x = - \frac{a c + a b + 2 + \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

$x = - \frac{a c + a b + 2 - \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

$x = - \frac{a c + a b + 2 + \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

Step 8

The final solution is all the values that make $x (a x^{2} + a x c + a x b + a b c - 2 x^{2} + 2 x + 24) = 0$ true.

$x = 0$

$x = - \frac{a c + a b + 2 - \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$

$x = - \frac{a c + a b + 2 + \sqrt{a^{2} c^{2} - 2 a^{2} b c + 4 a c + a^{2} b^{2} + 4 a b - 96 a + 8 a b c + 196}}{2 (a - 2)}$