Algebra Examples

$\log ({(x - 1)}^{2}) = \log (- 5 x - 1)$

Step 1

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

${(x - 1)}^{2} = - 5 x - 1$

Step 2

Solve for $x$ .

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Step 2.1

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

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Step 2.1.1

Add $5 x$ to both sides of the equation.

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

Step 2.1.2

Simplify each term.

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Step 2.1.2.1

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

$(x - 1) (x - 1) + 5 x = - 1$

Step 2.1.2.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 2.1.2.2.1

Apply the distributive property.

$x (x - 1) - 1 (x - 1) + 5 x = - 1$

Step 2.1.2.2.2

Apply the distributive property.

$x \cdot x + x \cdot - 1 - 1 (x - 1) + 5 x = - 1$

Step 2.1.2.2.3

Apply the distributive property.

$x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 + 5 x = - 1$

Step 2.1.2.3

Simplify and combine like terms.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1 + 5 x = - 1$

Step 2.1.2.3.1.2

Move $- 1$ to the left of $x$ .

$x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1 + 5 x = - 1$

Step 2.1.2.3.1.3

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x - 1 x - 1 \cdot - 1 + 5 x = - 1$

Step 2.1.2.3.1.4

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x - x - 1 \cdot - 1 + 5 x = - 1$

Step 2.1.2.3.1.5

Multiply $- 1$ by $- 1$ .

$x^{2} - x - x + 1 + 5 x = - 1$

Step 2.1.2.3.2

Subtract $x$ from $- x$ .

$x^{2} - 2 x + 1 + 5 x = - 1$

Step 2.1.3

Add $- 2 x$ and $5 x$ .

$x^{2} + 3 x + 1 = - 1$

Step 2.2

Add $1$ to both sides of the equation.

$x^{2} + 3 x + 1 + 1 = 0$

Step 2.3

Add $1$ and $1$ .

$x^{2} + 3 x + 2 = 0$

Step 2.4

Factor $x^{2} + 3 x + 2$ using the AC method.

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Step 2.4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $3$ .

$1, 2$

Step 2.4.2

Write the factored form using these integers.

$(x + 1) (x + 2) = 0$

Step 2.5

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

$x + 1 = 0$

$x + 2 = 0$

Step 2.6

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.6.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.7

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 2.7.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 2.7.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.8

The final solution is all the values that make $(x + 1) (x + 2) = 0$ true.

$x = - 1, - 2$