Algebra Examples

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

Step 1

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

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Step 1.1

Rewrite.

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

Step 1.2

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

$x - 1 = (2 - x) (2 - x)$

Step 1.3

Expand $(2 - x) (2 - x)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$x - 1 = 2 (2 - x) - x (2 - x)$

Step 1.3.2

Apply the distributive property.

$x - 1 = 2 \cdot 2 + 2 (- x) - x (2 - x)$

Step 1.3.3

Apply the distributive property.

$x - 1 = 2 \cdot 2 + 2 (- x) - x \cdot 2 - x (- x)$

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 $2$ by $2$ .

$x - 1 = 4 + 2 (- x) - x \cdot 2 - x (- x)$

Step 1.4.1.2

Multiply $- 1$ by $2$ .

$x - 1 = 4 - 2 x - x \cdot 2 - x (- x)$

Step 1.4.1.3

Multiply $2$ by $- 1$ .

$x - 1 = 4 - 2 x - 2 x - x (- x)$

Step 1.4.1.4

Rewrite using the commutative property of multiplication.

$x - 1 = 4 - 2 x - 2 x - 1 \cdot - 1 x \cdot x$

Step 1.4.1.5

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

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Step 1.4.1.5.1

Move $x$ .

$x - 1 = 4 - 2 x - 2 x - 1 \cdot - 1 (x \cdot x)$

Step 1.4.1.5.2

Multiply $x$ by $x$ .

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

Step 1.4.1.6

Multiply $- 1$ by $- 1$ .

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

Step 1.4.1.7

Multiply $x^{2}$ by $1$ .

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

Step 1.4.2

Subtract $2 x$ from $- 2 x$ .

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

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.

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

Step 3

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

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Step 3.1

Subtract $x$ from both sides of the equation.

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

Step 3.2

Subtract $x$ from $- 4 x$ .

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

Step 4

Move all terms to the left side of the equation and simplify.

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Step 4.1

Add $1$ to both sides of the equation.

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

Step 4.2

Add $4$ and $1$ .

$x^{2} - 5 x + 5 = 0$

Step 5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6

Substitute the values $a = 1$ , $b = - 5$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (1 \cdot 5)}}{2 \cdot 1}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot 5$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 5}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $5$ .

$x = \frac{5 \pm \sqrt{25 - 20}}{2 \cdot 1}$

Step 7.1.3

Subtract $20$ from $25$ .

$x = \frac{5 \pm \sqrt{5}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{5 \pm \sqrt{5}}{2}$

Step 8

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{5}}{2}, \frac{5 - \sqrt{5}}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 + \sqrt{5}}{2}, \frac{5 - \sqrt{5}}{2}$

Decimal Form:

$x = 3.61803398 \dots, 1.38196601 \dots$