Algebra Examples

$5 - \sqrt{26}$ , $5 + \sqrt{26}$

Step 1

$x = 5 - \sqrt{26}$ and $x = 5 + \sqrt{26}$ are the two real distinct solutions for the quadratic equation, which means that $x + 5 - \sqrt{26}$ and $x - 5 + \sqrt{26}$ are the factors of the quadratic equation.

$(x + 5 - \sqrt{26}) (x - 5 + \sqrt{26}) = 0$

Step 2

Expand $(x + 5 - \sqrt{26}) (x - 5 + \sqrt{26})$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x + x \cdot - 5 + x \sqrt{26} + 5 x + 5 \cdot - 5 + 5 \sqrt{26} - \sqrt{26} x - \sqrt{26} \cdot - 5 - \sqrt{26} \sqrt{26} = 0$

Step 3

Simplify terms.

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Step 3.1

Combine the opposite terms in $x \cdot x + x \cdot - 5 + x \sqrt{26} + 5 x + 5 \cdot - 5 + 5 \sqrt{26} - \sqrt{26} x - \sqrt{26} \cdot - 5 - \sqrt{26} \sqrt{26}$ .

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Step 3.1.1

Reorder the factors in the terms $x \cdot - 5$ and $5 x$ .

$x \cdot x - 5 x + x \sqrt{26} + 5 x + 5 \cdot - 5 + 5 \sqrt{26} - \sqrt{26} x - \sqrt{26} \cdot - 5 - \sqrt{26} \sqrt{26} = 0$

Step 3.1.2

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

$x \cdot x + x \sqrt{26} + 0 + 5 \cdot - 5 + 5 \sqrt{26} - \sqrt{26} x - \sqrt{26} \cdot - 5 - \sqrt{26} \sqrt{26} = 0$

Step 3.1.3

Add $x \cdot x + x \sqrt{26}$ and $0$ .

$x \cdot x + x \sqrt{26} + 5 \cdot - 5 + 5 \sqrt{26} - \sqrt{26} x - \sqrt{26} \cdot - 5 - \sqrt{26} \sqrt{26} = 0$

Step 3.1.4

Reorder the factors in the terms $x \sqrt{26}$ and $- \sqrt{26} x$ .

$x \cdot x + x \sqrt{26} + 5 \cdot - 5 + 5 \sqrt{26} - x \sqrt{26} - \sqrt{26} \cdot - 5 - \sqrt{26} \sqrt{26} = 0$

Step 3.1.5

Subtract $x \sqrt{26}$ from $x \sqrt{26}$ .

$x \cdot x + 0 + 5 \cdot - 5 + 5 \sqrt{26} - \sqrt{26} \cdot - 5 - \sqrt{26} \sqrt{26} = 0$

Step 3.1.6

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

$x \cdot x + 5 \cdot - 5 + 5 \sqrt{26} - \sqrt{26} \cdot - 5 - \sqrt{26} \sqrt{26} = 0$

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $x$ by $x$ .

$x^{2} + 5 \cdot - 5 + 5 \sqrt{26} - \sqrt{26} \cdot - 5 - \sqrt{26} \sqrt{26} = 0$

Step 3.2.2

Multiply $5$ by $- 5$ .

$x^{2} - 25 + 5 \sqrt{26} - \sqrt{26} \cdot - 5 - \sqrt{26} \sqrt{26} = 0$

Step 3.2.3

Multiply $- 5$ by $- 1$ .

$x^{2} - 25 + 5 \sqrt{26} + 5 \sqrt{26} - \sqrt{26} \sqrt{26} = 0$

Step 3.2.4

Multiply $- \sqrt{26} \sqrt{26}$ .

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Step 3.2.4.1

Raise $\sqrt{26}$ to the power of $1$ .

$x^{2} - 25 + 5 \sqrt{26} + 5 \sqrt{26} - (\sqrt{26} \sqrt{26}) = 0$

Step 3.2.4.2

Raise $\sqrt{26}$ to the power of $1$ .

$x^{2} - 25 + 5 \sqrt{26} + 5 \sqrt{26} - (\sqrt{26} \sqrt{26}) = 0$

Step 3.2.4.3

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

$x^{2} - 25 + 5 \sqrt{26} + 5 \sqrt{26} - {\sqrt{26}}^{1 + 1} = 0$

Step 3.2.4.4

Add $1$ and $1$ .

$x^{2} - 25 + 5 \sqrt{26} + 5 \sqrt{26} - {\sqrt{26}}^{2} = 0$

Step 3.2.5

Rewrite ${\sqrt{26}}^{2}$ as $26$ .

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Step 3.2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{26}$ as $26^{\frac{1}{2}}$ .

$x^{2} - 25 + 5 \sqrt{26} + 5 \sqrt{26} - {(26^{\frac{1}{2}})}^{2} = 0$

Step 3.2.5.2

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

$x^{2} - 25 + 5 \sqrt{26} + 5 \sqrt{26} - 26^{\frac{1}{2} \cdot 2} = 0$

Step 3.2.5.3

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

$x^{2} - 25 + 5 \sqrt{26} + 5 \sqrt{26} - 26^{\frac{2}{2}} = 0$

Step 3.2.5.4

Cancel the common factor of $2$ .

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Step 3.2.5.4.1

Cancel the common factor.

$x^{2} - 25 + 5 \sqrt{26} + 5 \sqrt{26} - 26^{\frac{2}{2}} = 0$

Step 3.2.5.4.2

Rewrite the expression.

$x^{2} - 25 + 5 \sqrt{26} + 5 \sqrt{26} - 26 = 0$

Step 3.2.5.5

Evaluate the exponent.

$x^{2} - 25 + 5 \sqrt{26} + 5 \sqrt{26} - 1 \cdot 26 = 0$

Step 3.2.6

Multiply $- 1$ by $26$ .

$x^{2} - 25 + 5 \sqrt{26} + 5 \sqrt{26} - 26 = 0$

Step 3.3

Simplify by adding terms.

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Step 3.3.1

Subtract $26$ from $- 25$ .

$x^{2} - 51 + 5 \sqrt{26} + 5 \sqrt{26} = 0$

Step 3.3.2

Add $5 \sqrt{26}$ and $5 \sqrt{26}$ .

$x^{2} - 51 + 10 \sqrt{26} = 0$

Step 4

The standard quadratic equation using the given set of solutions ${5 - \sqrt{26}, 5 + \sqrt{26}}$ is $y = x^{2} - 51 + 10 \sqrt{26}$ .

$y = x^{2} - 51 + 10 \sqrt{26}$

Step 5