Algebra Examples

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

Step 1

Add $1$ to both sides of the equation.

$x^{2} + x = 1$

Step 2

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(\frac{1}{2})}^{2}$

Step 3

Add the term to each side of the equation.

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

Step 4

Simplify the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 4.1.1.2

One to any power is one.

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

Step 4.1.1.3

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

$x^{2} + x + \frac{1}{4} = 1 + {(\frac{1}{2})}^{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $1 + {(\frac{1}{2})}^{2}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the product rule to $\frac{1}{2}$ .

$x^{2} + x + \frac{1}{4} = 1 + \frac{1^{2}}{2^{2}}$

Step 4.2.1.1.2

One to any power is one.

$x^{2} + x + \frac{1}{4} = 1 + \frac{1}{2^{2}}$

Step 4.2.1.1.3

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

$x^{2} + x + \frac{1}{4} = 1 + \frac{1}{4}$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Write $1$ as a fraction with a common denominator.

$x^{2} + x + \frac{1}{4} = \frac{4}{4} + \frac{1}{4}$

Step 4.2.1.2.2

Combine the numerators over the common denominator.

$x^{2} + x + \frac{1}{4} = \frac{4 + 1}{4}$

Step 4.2.1.2.3

Add $4$ and $1$ .

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

Step 5

Factor the perfect trinomial square into ${(x + \frac{1}{2})}^{2}$ .

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

Step 6

Solve the equation for $x$ .

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Step 6.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x + \frac{1}{2} = \pm \sqrt{\frac{5}{4}}$

Step 6.2

Simplify $\pm \sqrt{\frac{5}{4}}$ .

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Step 6.2.1

Rewrite $\sqrt{\frac{5}{4}}$ as $\frac{\sqrt{5}}{\sqrt{4}}$ .

$x + \frac{1}{2} = \pm \frac{\sqrt{5}}{\sqrt{4}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Rewrite $4$ as $2^{2}$ .

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

Step 6.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

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

Step 6.3

Subtract $\frac{1}{2}$ from both sides of the equation.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.61803398 \dots, - 1.61803398 \dots$