Algebra Examples

$\sqrt{5}$ , $- \sqrt{5}$

Step 1

$x = \sqrt{5}$ and $x = - \sqrt{5}$ are the two real distinct solutions for the quadratic equation, which means that $x - \sqrt{5}$ and $x + \sqrt{5}$ are the factors of the quadratic equation.

$(x - \sqrt{5}) (x + \sqrt{5}) = 0$

Step 2

Expand $(x - \sqrt{5}) (x + \sqrt{5})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$x (x + \sqrt{5}) - \sqrt{5} (x + \sqrt{5}) = 0$

Step 2.2

Apply the distributive property.

$x \cdot x + x \sqrt{5} - \sqrt{5} (x + \sqrt{5}) = 0$

Step 2.3

Apply the distributive property.

$x \cdot x + x \sqrt{5} - \sqrt{5} x - \sqrt{5} \sqrt{5} = 0$

Step 3

Simplify terms.

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Step 3.1

Combine the opposite terms in $x \cdot x + x \sqrt{5} - \sqrt{5} x - \sqrt{5} \sqrt{5}$ .

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Step 3.1.1

Reorder the factors in the terms $x \sqrt{5}$ and $- \sqrt{5} x$ .

$x \cdot x + x \sqrt{5} - x \sqrt{5} - \sqrt{5} \sqrt{5} = 0$

Step 3.1.2

Subtract $x \sqrt{5}$ from $x \sqrt{5}$ .

$x \cdot x + 0 - \sqrt{5} \sqrt{5} = 0$

Step 3.1.3

Add $x \cdot x$ and $0$ .

$x \cdot x - \sqrt{5} \sqrt{5} = 0$

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $x$ by $x$ .

$x^{2} - \sqrt{5} \sqrt{5} = 0$

Step 3.2.2

Multiply $- \sqrt{5} \sqrt{5}$ .

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Step 3.2.2.1

Raise $\sqrt{5}$ to the power of $1$ .

$x^{2} - (\sqrt{5} \sqrt{5}) = 0$

Step 3.2.2.2

Raise $\sqrt{5}$ to the power of $1$ .

$x^{2} - (\sqrt{5} \sqrt{5}) = 0$

Step 3.2.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.2.2.4

Add $1$ and $1$ .

$x^{2} - {\sqrt{5}}^{2} = 0$

Step 3.2.3

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$x^{2} - {(5^{\frac{1}{2}})}^{2} = 0$

Step 3.2.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{2} - 5^{\frac{1}{2} \cdot 2} = 0$

Step 3.2.3.3

Combine $\frac{1}{2}$ and $2$ .

$x^{2} - 5^{\frac{2}{2}} = 0$

Step 3.2.3.4

Cancel the common factor of $2$ .

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Step 3.2.3.4.1

Cancel the common factor.

$x^{2} - 5^{\frac{2}{2}} = 0$

Step 3.2.3.4.2

Rewrite the expression.

$x^{2} - 5 = 0$

Step 3.2.3.5

Evaluate the exponent.

$x^{2} - 1 \cdot 5 = 0$

Step 3.2.4

Multiply $- 1$ by $5$ .

$x^{2} - 5 = 0$

Step 4

The standard quadratic equation using the given set of solutions ${\sqrt{5}, - \sqrt{5}}$ is $y = x^{2} - 5$ .

$y = x^{2} - 5$

Step 5