Algebra Examples

${(x + 4)}^{2} - 2 x - 5 = {(x - 1)}^{2}$

Step 1

Simplify ${(x - 1)}^{2}$ .

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Step 1.1

Rewrite.

${(x + 4)}^{2} - 2 x - 5 = 0 + 0 + {(x - 1)}^{2}$

Step 1.2

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

${(x + 4)}^{2} - 2 x - 5 = (x - 1) (x - 1)$

Step 1.3

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

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Step 1.3.1

Apply the distributive property.

${(x + 4)}^{2} - 2 x - 5 = x (x - 1) - 1 (x - 1)$

Step 1.3.2

Apply the distributive property.

${(x + 4)}^{2} - 2 x - 5 = x \cdot x + x \cdot - 1 - 1 (x - 1)$

Step 1.3.3

Apply the distributive property.

${(x + 4)}^{2} - 2 x - 5 = x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $x$ by $x$ .

${(x + 4)}^{2} - 2 x - 5 = x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1$

Step 1.4.1.2

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

${(x + 4)}^{2} - 2 x - 5 = x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1$

Step 1.4.1.3

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

${(x + 4)}^{2} - 2 x - 5 = x^{2} - x - 1 x - 1 \cdot - 1$

Step 1.4.1.4

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

${(x + 4)}^{2} - 2 x - 5 = x^{2} - x - x - 1 \cdot - 1$

Step 1.4.1.5

Multiply $- 1$ by $- 1$ .

${(x + 4)}^{2} - 2 x - 5 = x^{2} - x - x + 1$

Step 1.4.2

Subtract $x$ from $- x$ .

${(x + 4)}^{2} - 2 x - 5 = x^{2} - 2 x + 1$

Step 2

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

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Step 2.1

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

${(x + 4)}^{2} - 2 x - 5 - x^{2} = - 2 x + 1$

Step 2.2

Add $2 x$ to both sides of the equation.

${(x + 4)}^{2} - 2 x - 5 - x^{2} + 2 x = 1$

Step 2.3

Combine the opposite terms in ${(x + 4)}^{2} - 2 x - 5 - x^{2} + 2 x$ .

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Step 2.3.1

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

${(x + 4)}^{2} - 5 - x^{2} + 0 = 1$

Step 2.3.2

Add ${(x + 4)}^{2} - 5 - x^{2}$ and $0$ .

${(x + 4)}^{2} - 5 - x^{2} = 1$

Step 2.4

Simplify each term.

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Step 2.4.1

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$(x + 4) (x + 4) - 5 - x^{2} = 1$

Step 2.4.2

Expand $(x + 4) (x + 4)$ using the FOIL Method.

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Step 2.4.2.1

Apply the distributive property.

$x (x + 4) + 4 (x + 4) - 5 - x^{2} = 1$

Step 2.4.2.2

Apply the distributive property.

$x \cdot x + x \cdot 4 + 4 (x + 4) - 5 - x^{2} = 1$

Step 2.4.2.3

Apply the distributive property.

$x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4 - 5 - x^{2} = 1$

Step 2.4.3

Simplify and combine like terms.

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 4 + 4 x + 4 \cdot 4 - 5 - x^{2} = 1$

Step 2.4.3.1.2

Move $4$ to the left of $x$ .

$x^{2} + 4 \cdot x + 4 x + 4 \cdot 4 - 5 - x^{2} = 1$

Step 2.4.3.1.3

Multiply $4$ by $4$ .

$x^{2} + 4 x + 4 x + 16 - 5 - x^{2} = 1$

Step 2.4.3.2

Add $4 x$ and $4 x$ .

$x^{2} + 8 x + 16 - 5 - x^{2} = 1$

Step 2.5

Combine the opposite terms in $x^{2} + 8 x + 16 - 5 - x^{2}$ .

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Step 2.5.1

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

$8 x + 16 - 5 + 0 = 1$

Step 2.5.2

Add $8 x + 16 - 5$ and $0$ .

$8 x + 16 - 5 = 1$

Step 2.6

Subtract $5$ from $16$ .

$8 x + 11 = 1$

Step 3

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

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Step 3.1

Subtract $11$ from both sides of the equation.

$8 x = 1 - 11$

Step 3.2

Subtract $11$ from $1$ .

$8 x = - 10$

Step 4

Divide each term in $8 x = - 10$ by $8$ and simplify.

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Step 4.1

Divide each term in $8 x = - 10$ by $8$ .

$\frac{8 x}{8} = \frac{- 10}{8}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $8$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 10}{8}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 10}{8}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Cancel the common factor of $- 10$ and $8$ .

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Step 4.3.1.1

Factor $2$ out of $- 10$ .

$x = \frac{2 (- 5)}{8}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $2$ out of $8$ .

$x = \frac{2 \cdot - 5}{2 \cdot 4}$

Step 4.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot - 5}{2 \cdot 4}$

Step 4.3.1.2.3

Rewrite the expression.

$x = \frac{- 5}{4}$

Step 4.3.2

Move the negative in front of the fraction.

$x = - \frac{5}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{5}{4}$

Decimal Form:

$x = - 1.25$

Mixed Number Form:

$x = - 1 \frac{1}{4}$