Algebra Examples

$(a x + 2) (b x + 7) = 15 x^{2} + c x + 14$

Step 1

Rewrite the equation as $15 x^{2} + c x + 14 = (a x + 2) (b x + 7)$ .

$15 x^{2} + c x + 14 = (a x + 2) (b x + 7)$

Step 2

Simplify $(a x + 2) (b x + 7)$ .

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Step 2.1

Rewrite.

$15 x^{2} + c x + 14 = 0 + 0 + (a x + 2) (b x + 7)$

Step 2.2

Simplify by adding zeros.

$15 x^{2} + c x + 14 = (a x + 2) (b x + 7)$

Step 2.3

Expand $(a x + 2) (b x + 7)$ using the FOIL Method.

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Step 2.3.1

Apply the distributive property.

$15 x^{2} + c x + 14 = a x (b x + 7) + 2 (b x + 7)$

Step 2.3.2

Apply the distributive property.

$15 x^{2} + c x + 14 = a x (b x) + a x \cdot 7 + 2 (b x + 7)$

Step 2.3.3

Apply the distributive property.

$15 x^{2} + c x + 14 = a x (b x) + a x \cdot 7 + 2 (b x) + 2 \cdot 7$

Step 2.4

Simplify each term.

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Step 2.4.1

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

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Step 2.4.1.1

Move $x$ .

$15 x^{2} + c x + 14 = a (x \cdot x) b + a x \cdot 7 + 2 (b x) + 2 \cdot 7$

Step 2.4.1.2

Multiply $x$ by $x$ .

$15 x^{2} + c x + 14 = a x^{2} b + a x \cdot 7 + 2 (b x) + 2 \cdot 7$

Step 2.4.2

Move $7$ to the left of $a x$ .

$15 x^{2} + c x + 14 = a x^{2} b + 7 \cdot (a x) + 2 (b x) + 2 \cdot 7$

Step 2.4.3

Multiply $2$ by $7$ .

$15 x^{2} + c x + 14 = a x^{2} b + 7 a x + 2 b x + 14$

Step 3

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

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Step 3.1

Subtract $15 x^{2}$ from both sides of the equation.

$c x + 14 = a x^{2} b + 7 a x + 2 b x + 14 - 15 x^{2}$

Step 3.2

Subtract $14$ from both sides of the equation.

$c x = a x^{2} b + 7 a x + 2 b x + 14 - 15 x^{2} - 14$

Step 3.3

Combine the opposite terms in $a x^{2} b + 7 a x + 2 b x + 14 - 15 x^{2} - 14$ .

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Step 3.3.1

Subtract $14$ from $14$ .

$c x = a x^{2} b + 7 a x + 2 b x - 15 x^{2} + 0$

Step 3.3.2

Add $a x^{2} b + 7 a x + 2 b x - 15 x^{2}$ and $0$ .

$c x = a x^{2} b + 7 a x + 2 b x - 15 x^{2}$

Step 4

Divide each term in $c x = a x^{2} b + 7 a x + 2 b x - 15 x^{2}$ by $x$ and simplify.

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Step 4.1

Divide each term in $c x = a x^{2} b + 7 a x + 2 b x - 15 x^{2}$ by $x$ .

$\frac{c x}{x} = \frac{a x^{2} b}{x} + \frac{7 a x}{x} + \frac{2 b x}{x} + \frac{- 15 x^{2}}{x}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $x$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{c x}{x} = \frac{a x^{2} b}{x} + \frac{7 a x}{x} + \frac{2 b x}{x} + \frac{- 15 x^{2}}{x}$

Step 4.2.1.2

Divide $c$ by $1$ .

$c = \frac{a x^{2} b}{x} + \frac{7 a x}{x} + \frac{2 b x}{x} + \frac{- 15 x^{2}}{x}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Cancel the common factor of $x^{2}$ and $x$ .

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Step 4.3.1.1.1

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

$c = \frac{x (a x b)}{x} + \frac{7 a x}{x} + \frac{2 b x}{x} + \frac{- 15 x^{2}}{x}$

Step 4.3.1.1.2

Cancel the common factors.

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Step 4.3.1.1.2.1

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

$c = \frac{x (a x b)}{x^{1}} + \frac{7 a x}{x} + \frac{2 b x}{x} + \frac{- 15 x^{2}}{x}$

Step 4.3.1.1.2.2

Factor $x$ out of $x^{1}$ .

$c = \frac{x (a x b)}{x \cdot 1} + \frac{7 a x}{x} + \frac{2 b x}{x} + \frac{- 15 x^{2}}{x}$

Step 4.3.1.1.2.3

Cancel the common factor.

$c = \frac{x (a x b)}{x \cdot 1} + \frac{7 a x}{x} + \frac{2 b x}{x} + \frac{- 15 x^{2}}{x}$

Step 4.3.1.1.2.4

Rewrite the expression.

$c = \frac{a x b}{1} + \frac{7 a x}{x} + \frac{2 b x}{x} + \frac{- 15 x^{2}}{x}$

Step 4.3.1.1.2.5

Divide $a x b$ by $1$ .

$c = a x b + \frac{7 a x}{x} + \frac{2 b x}{x} + \frac{- 15 x^{2}}{x}$

Step 4.3.1.2

Cancel the common factor of $x$ .

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Step 4.3.1.2.1

Cancel the common factor.

$c = a x b + \frac{7 a x}{x} + \frac{2 b x}{x} + \frac{- 15 x^{2}}{x}$

Step 4.3.1.2.2

Divide $7 a$ by $1$ .

$c = a x b + 7 a + \frac{2 b x}{x} + \frac{- 15 x^{2}}{x}$

Step 4.3.1.3

Cancel the common factor of $x$ .

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Step 4.3.1.3.1

Cancel the common factor.

$c = a x b + 7 a + \frac{2 b x}{x} + \frac{- 15 x^{2}}{x}$

Step 4.3.1.3.2

Divide $2 b$ by $1$ .

$c = a x b + 7 a + 2 b + \frac{- 15 x^{2}}{x}$

Step 4.3.1.4

Cancel the common factor of $x^{2}$ and $x$ .

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Step 4.3.1.4.1

Factor $x$ out of $- 15 x^{2}$ .

$c = a x b + 7 a + 2 b + \frac{x (- 15 x)}{x}$

Step 4.3.1.4.2

Cancel the common factors.

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Step 4.3.1.4.2.1

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

$c = a x b + 7 a + 2 b + \frac{x (- 15 x)}{x^{1}}$

Step 4.3.1.4.2.2

Factor $x$ out of $x^{1}$ .

$c = a x b + 7 a + 2 b + \frac{x (- 15 x)}{x \cdot 1}$

Step 4.3.1.4.2.3

Cancel the common factor.

$c = a x b + 7 a + 2 b + \frac{x (- 15 x)}{x \cdot 1}$

Step 4.3.1.4.2.4

Rewrite the expression.

$c = a x b + 7 a + 2 b + \frac{- 15 x}{1}$

Step 4.3.1.4.2.5

Divide $- 15 x$ by $1$ .

$c = a x b + 7 a + 2 b - 15 x$