Precalculus Examples

$a^{2} x + (a - 1) = (a + 1) x$

Step 1

Simplify $(a + 1) x$ .

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Step 1.1

Rewrite.

$a^{2} x + a - 1 = 0 + 0 + (a + 1) x$

Step 1.2

Simplify by adding zeros.

$a^{2} x + a - 1 = (a + 1) x$

Step 1.3

Apply the distributive property.

$a^{2} x + a - 1 = a x + 1 x$

Step 1.4

Multiply $x$ by $1$ .

$a^{2} x + a - 1 = a x + x$

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.

$a x + x = a^{2} x + a - 1$

Step 3

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

$a x + x - a^{2} x = a - 1$

Step 4

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

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Step 4.1

Factor $x$ out of $a x$ .

$x a + x - a^{2} x = a - 1$

Step 4.2

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

$x a + x - a^{2} x = a - 1$

Step 4.3

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

$x a + x \cdot 1 - a^{2} x = a - 1$

Step 4.4

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

$x a + x \cdot 1 + x (- a^{2}) = a - 1$

Step 4.5

Factor $x$ out of $x a + x \cdot 1$ .

$x (a + 1) + x (- a^{2}) = a - 1$

Step 4.6

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

$x (a + 1 - a^{2}) = a - 1$

Step 5

Reorder terms.

$x (- a^{2} + a + 1) = a - 1$

Step 6

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

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Step 6.1

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

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

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $- a^{2} + a + 1$ .

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{a}{- a^{2} + a + 1} + \frac{- 1}{- a^{2} + a + 1}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Combine the numerators over the common denominator.

$x = \frac{a - 1}{- a^{2} + a + 1}$

Step 6.3.2

Factor $- 1$ out of $- a^{2}$ .

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

Step 6.3.3

Factor $- 1$ out of $a$ .

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

Step 6.3.4

Factor $- 1$ out of $- (a^{2}) - 1 (- a)$ .

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

Step 6.3.5

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

$x = \frac{a - 1}{- (a^{2} - a) - 1 (- 1)}$

Step 6.3.6

Factor $- 1$ out of $- (a^{2} - a) - 1 (- 1)$ .

$x = \frac{a - 1}{- (a^{2} - a - 1)}$

Step 6.3.7

Rewrite negatives.

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Step 6.3.7.1

Rewrite $- (a^{2} - a - 1)$ as $- 1 (a^{2} - a - 1)$ .

$x = \frac{a - 1}{- 1 (a^{2} - a - 1)}$

Step 6.3.7.2

Move the negative in front of the fraction.

$x = - \frac{a - 1}{a^{2} - a - 1}$