Algebra Examples

$(8 x^{2} - 15 x) - (x^{2} - 27 x) = a x^{2} + b x$

Step 1

Simplify $8 x^{2} - 15 x - (x^{2} - 27 x)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$8 x^{2} - 15 x - x^{2} - (- 27 x) = a x^{2} + b x$

Step 1.1.2

Multiply $- 27$ by $- 1$ .

$8 x^{2} - 15 x - x^{2} + 27 x = a x^{2} + b x$

Step 1.2

Simplify by adding terms.

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Step 1.2.1

Subtract $x^{2}$ from $8 x^{2}$ .

$7 x^{2} - 15 x + 27 x = a x^{2} + b x$

Step 1.2.2

Add $- 15 x$ and $27 x$ .

$7 x^{2} + 12 x = a x^{2} + b x$

Step 2

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

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Step 2.1

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

$7 x^{2} + 12 x - a x^{2} = b x$

Step 2.2

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

$7 x^{2} + 12 x - a x^{2} - b x = 0$

Step 3

Factor $x$ out of $7 x^{2} + 12 x - a x^{2} - b x$ .

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Step 3.1

Factor $x$ out of $7 x^{2}$ .

$x (7 x) + 12 x - a x^{2} - b x = 0$

Step 3.2

Factor $x$ out of $12 x$ .

$x (7 x) + x \cdot 12 - a x^{2} - b x = 0$

Step 3.3

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

$x (7 x) + x \cdot 12 + x (- a x) - b x = 0$

Step 3.4

Factor $x$ out of $- b x$ .

$x (7 x) + x \cdot 12 + x (- a x) + x (- b) = 0$

Step 3.5

Factor $x$ out of $x (7 x) + x \cdot 12$ .

$x (7 x + 12) + x (- a x) + x (- b) = 0$

Step 3.6

Factor $x$ out of $x (7 x + 12) + x (- a x)$ .

$x (7 x + 12 - a x) + x (- b) = 0$

Step 3.7

Factor $x$ out of $x (7 x + 12 - a x) + x (- b)$ .

$x (7 x + 12 - a x - b) = 0$

Step 4

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

$x = 0$

$7 x + 12 - a x - b = 0$

Step 5

Set $x$ equal to $0$ .

$x = 0$

Step 6

Set $7 x + 12 - a x - b$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $7 x + 12 - a x - b$ equal to $0$ .

$7 x + 12 - a x - b = 0$

Step 6.2

Solve $7 x + 12 - a x - b = 0$ for $x$ .

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Step 6.2.1

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

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Step 6.2.1.1

Subtract $12$ from both sides of the equation.

$7 x - a x - b = - 12$

Step 6.2.1.2

Add $b$ to both sides of the equation.

$7 x - a x = - 12 + b$

Step 6.2.2

Factor $x$ out of $7 x - a x$ .

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Step 6.2.2.1

Factor $x$ out of $7 x$ .

$x \cdot 7 - a x = - 12 + b$

Step 6.2.2.2

Factor $x$ out of $- a x$ .

$x \cdot 7 + x (- a) = - 12 + b$

Step 6.2.2.3

Factor $x$ out of $x \cdot 7 + x (- a)$ .

$x (7 - a) = - 12 + b$

Step 6.2.3

Divide each term in $x (7 - a) = - 12 + b$ by $7 - a$ and simplify.

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Step 6.2.3.1

Divide each term in $x (7 - a) = - 12 + b$ by $7 - a$ .

$\frac{x (7 - a)}{7 - a} = \frac{- 12}{7 - a} + \frac{b}{7 - a}$

Step 6.2.3.2

Simplify the left side.

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Step 6.2.3.2.1

Cancel the common factor of $7 - a$ .

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Step 6.2.3.2.1.1

Cancel the common factor.

$\frac{x (7 - a)}{7 - a} = \frac{- 12}{7 - a} + \frac{b}{7 - a}$

Step 6.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 12}{7 - a} + \frac{b}{7 - a}$

Step 6.2.3.3

Simplify the right side.

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Step 6.2.3.3.1

Move the negative in front of the fraction.

$x = - \frac{12}{7 - a} + \frac{b}{7 - a}$

Step 6.2.3.3.2

Combine the numerators over the common denominator.

$x = \frac{- 12 + b}{7 - a}$

Step 6.2.3.3.3

Rewrite $- 12$ as $- 1 (12)$ .

$x = \frac{- 1 (12) + b}{7 - a}$

Step 6.2.3.3.4

Factor $- 1$ out of $b$ .

$x = \frac{- 1 (12) - 1 (- b)}{7 - a}$

Step 6.2.3.3.5

Factor $- 1$ out of $- 1 (12) - 1 (- b)$ .

$x = \frac{- 1 (12 - b)}{7 - a}$

Step 6.2.3.3.6

Move the negative in front of the fraction.

$x = - \frac{12 - b}{7 - a}$

Step 7

The final solution is all the values that make $x (7 x + 12 - a x - b) = 0$ true.

$x = 0$

$x = - \frac{12 - b}{7 - a}$