Algebra Examples

$3 - \frac{\sqrt{6}}{3}$ , $3 + \frac{\sqrt{6}}{3}$

Step 1

$x = 3 - \frac{\sqrt{6}}{3}$ and $x = 3 + \frac{\sqrt{6}}{3}$ are the two real distinct solutions for the quadratic equation, which means that $x - 3 - \frac{\sqrt{6}}{3}$ and $x - 3 + \frac{\sqrt{6}}{3}$ are the factors of the quadratic equation.

$(x - 3 - \frac{\sqrt{6}}{3}) (x - 3 + \frac{\sqrt{6}}{3}) = 0$

Step 2

Expand $(x - 3 - \frac{\sqrt{6}}{3}) (x - 3 + \frac{\sqrt{6}}{3})$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x + x \cdot - 3 + x (\frac{\sqrt{6}}{3}) - 3 x - 3 \cdot - 3 - 3 \frac{\sqrt{6}}{3} - \frac{\sqrt{6}}{3} x - \frac{\sqrt{6}}{3} \cdot - 3 - \frac{\sqrt{6}}{3} \cdot \frac{\sqrt{6}}{3} = 0$

Step 3

Simplify terms.

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Step 3.1

Combine the opposite terms in $x \cdot x + x \cdot - 3 + x \frac{\sqrt{6}}{3} - 3 x - 3 \cdot - 3 - 3 \frac{\sqrt{6}}{3} - \frac{\sqrt{6}}{3} x - \frac{\sqrt{6}}{3} \cdot - 3 - \frac{\sqrt{6}}{3} \cdot \frac{\sqrt{6}}{3}$ .

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Step 3.1.1

Reorder the factors in the terms $x \frac{\sqrt{6}}{3}$ and $- \frac{\sqrt{6}}{3} x$ .

$x \cdot x + x \cdot - 3 + \frac{\sqrt{6}}{3} x - 3 x - 3 \cdot - 3 - 3 \frac{\sqrt{6}}{3} - \frac{\sqrt{6}}{3} x - \frac{\sqrt{6}}{3} \cdot - 3 - \frac{\sqrt{6}}{3} \cdot \frac{\sqrt{6}}{3} = 0$

Step 3.1.2

Subtract $\frac{\sqrt{6}}{3} x$ from $\frac{\sqrt{6}}{3} x$ .

$x \cdot x + x \cdot - 3 + 0 - 3 x - 3 \cdot - 3 - 3 \frac{\sqrt{6}}{3} - \frac{\sqrt{6}}{3} \cdot - 3 - \frac{\sqrt{6}}{3} \cdot \frac{\sqrt{6}}{3} = 0$

Step 3.1.3

Add $x \cdot x + x \cdot - 3$ and $0$ .

$x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 - 3 \frac{\sqrt{6}}{3} - \frac{\sqrt{6}}{3} \cdot - 3 - \frac{\sqrt{6}}{3} \cdot \frac{\sqrt{6}}{3} = 0$

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3 - 3 \frac{\sqrt{6}}{3} - \frac{\sqrt{6}}{3} \cdot - 3 - \frac{\sqrt{6}}{3} \cdot \frac{\sqrt{6}}{3} = 0$

Step 3.2.2

Move $- 3$ to the left of $x$ .

$x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3 - 3 \frac{\sqrt{6}}{3} - \frac{\sqrt{6}}{3} \cdot - 3 - \frac{\sqrt{6}}{3} \cdot \frac{\sqrt{6}}{3} = 0$

Step 3.2.3

Multiply $- 3$ by $- 3$ .

$x^{2} - 3 x - 3 x + 9 - 3 \frac{\sqrt{6}}{3} - \frac{\sqrt{6}}{3} \cdot - 3 - \frac{\sqrt{6}}{3} \cdot \frac{\sqrt{6}}{3} = 0$

Step 3.2.4

Cancel the common factor of $3$ .

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Step 3.2.4.1

Factor $3$ out of $- 3$ .

$x^{2} - 3 x - 3 x + 9 + 3 (- 1) (\frac{\sqrt{6}}{3}) - \frac{\sqrt{6}}{3} \cdot - 3 - \frac{\sqrt{6}}{3} \cdot \frac{\sqrt{6}}{3} = 0$

Step 3.2.4.2

Cancel the common factor.

$x^{2} - 3 x - 3 x + 9 + 3 \cdot (- 1 \frac{\sqrt{6}}{3}) - \frac{\sqrt{6}}{3} \cdot - 3 - \frac{\sqrt{6}}{3} \cdot \frac{\sqrt{6}}{3} = 0$

Step 3.2.4.3

Rewrite the expression.

$x^{2} - 3 x - 3 x + 9 - 1 \sqrt{6} - \frac{\sqrt{6}}{3} \cdot - 3 - \frac{\sqrt{6}}{3} \cdot \frac{\sqrt{6}}{3} = 0$

Step 3.2.5

Rewrite $- 1 \sqrt{6}$ as $- \sqrt{6}$ .

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} - \frac{\sqrt{6}}{3} \cdot - 3 - \frac{\sqrt{6}}{3} \cdot \frac{\sqrt{6}}{3} = 0$

Step 3.2.6

Cancel the common factor of $3$ .

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Step 3.2.6.1

Move the leading negative in $- \frac{\sqrt{6}}{3}$ into the numerator.

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} + \frac{- \sqrt{6}}{3} \cdot - 3 - \frac{\sqrt{6}}{3} \cdot \frac{\sqrt{6}}{3} = 0$

Step 3.2.6.2

Factor $3$ out of $- 3$ .

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} + \frac{- \sqrt{6}}{3} \cdot (3 (- 1)) - \frac{\sqrt{6}}{3} \cdot \frac{\sqrt{6}}{3} = 0$

Step 3.2.6.3

Cancel the common factor.

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} + \frac{- \sqrt{6}}{3} \cdot (3 \cdot - 1) - \frac{\sqrt{6}}{3} \cdot \frac{\sqrt{6}}{3} = 0$

Step 3.2.6.4

Rewrite the expression.

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} - \sqrt{6} \cdot - 1 - \frac{\sqrt{6}}{3} \cdot \frac{\sqrt{6}}{3} = 0$

Step 3.2.7

Multiply $- 1$ by $- 1$ .

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} + 1 \sqrt{6} - \frac{\sqrt{6}}{3} \cdot \frac{\sqrt{6}}{3} = 0$

Step 3.2.8

Multiply $\sqrt{6}$ by $1$ .

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} + \sqrt{6} - \frac{\sqrt{6}}{3} \cdot \frac{\sqrt{6}}{3} = 0$

Step 3.2.9

Multiply $- \frac{\sqrt{6}}{3} \cdot \frac{\sqrt{6}}{3}$ .

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Step 3.2.9.1

Multiply $\frac{\sqrt{6}}{3}$ by $\frac{\sqrt{6}}{3}$ .

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} + \sqrt{6} - \frac{\sqrt{6} \sqrt{6}}{3 \cdot 3} = 0$

Step 3.2.9.2

Raise $\sqrt{6}$ to the power of $1$ .

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} + \sqrt{6} - \frac{\sqrt{6} \sqrt{6}}{3 \cdot 3} = 0$

Step 3.2.9.3

Raise $\sqrt{6}$ to the power of $1$ .

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} + \sqrt{6} - \frac{\sqrt{6} \sqrt{6}}{3 \cdot 3} = 0$

Step 3.2.9.4

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

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} + \sqrt{6} - \frac{{\sqrt{6}}^{1 + 1}}{3 \cdot 3} = 0$

Step 3.2.9.5

Add $1$ and $1$ .

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} + \sqrt{6} - \frac{{\sqrt{6}}^{2}}{3 \cdot 3} = 0$

Step 3.2.9.6

Multiply $3$ by $3$ .

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} + \sqrt{6} - \frac{{\sqrt{6}}^{2}}{9} = 0$

Step 3.2.10

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 3.2.10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} + \sqrt{6} - \frac{{(6^{\frac{1}{2}})}^{2}}{9} = 0$

Step 3.2.10.2

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

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} + \sqrt{6} - \frac{6^{\frac{1}{2} \cdot 2}}{9} = 0$

Step 3.2.10.3

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

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} + \sqrt{6} - \frac{6^{\frac{2}{2}}}{9} = 0$

Step 3.2.10.4

Cancel the common factor of $2$ .

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Step 3.2.10.4.1

Cancel the common factor.

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} + \sqrt{6} - \frac{6^{\frac{2}{2}}}{9} = 0$

Step 3.2.10.4.2

Rewrite the expression.

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} + \sqrt{6} - \frac{6}{9} = 0$

Step 3.2.10.5

Evaluate the exponent.

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} + \sqrt{6} - \frac{6}{9} = 0$

Step 3.2.11

Cancel the common factor of $6$ and $9$ .

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Step 3.2.11.1

Factor $3$ out of $6$ .

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} + \sqrt{6} - \frac{3 (2)}{9} = 0$

Step 3.2.11.2

Cancel the common factors.

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Step 3.2.11.2.1

Factor $3$ out of $9$ .

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} + \sqrt{6} - \frac{3 \cdot 2}{3 \cdot 3} = 0$

Step 3.2.11.2.2

Cancel the common factor.

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} + \sqrt{6} - \frac{3 \cdot 2}{3 \cdot 3} = 0$

Step 3.2.11.2.3

Rewrite the expression.

$x^{2} - 3 x - 3 x + 9 - \sqrt{6} + \sqrt{6} - \frac{2}{3} = 0$

Step 3.3

Simplify by adding terms.

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Step 3.3.1

Combine the opposite terms in $x^{2} - 3 x - 3 x + 9 - \sqrt{6} + \sqrt{6} - \frac{2}{3}$ .

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Step 3.3.1.1

Add $- \sqrt{6}$ and $\sqrt{6}$ .

$x^{2} - 3 x - 3 x + 9 + 0 - \frac{2}{3} = 0$

Step 3.3.1.2

Add $x^{2} - 3 x - 3 x + 9$ and $0$ .

$x^{2} - 3 x - 3 x + 9 - \frac{2}{3} = 0$

Step 3.3.2

Subtract $3 x$ from $- 3 x$ .

$x^{2} - 6 x + 9 - \frac{2}{3} = 0$

Step 4

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{2} - 6 x + 9 \cdot \frac{3}{3} - \frac{2}{3} = 0$

Step 5

Combine $9$ and $\frac{3}{3}$ .

$x^{2} - 6 x + \frac{9 \cdot 3}{3} - \frac{2}{3} = 0$

Step 6

Combine the numerators over the common denominator.

$x^{2} - 6 x + \frac{9 \cdot 3 - 2}{3} = 0$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $9$ by $3$ .

$x^{2} - 6 x + \frac{27 - 2}{3} = 0$

Step 7.2

Subtract $2$ from $27$ .

$x^{2} - 6 x + \frac{25}{3} = 0$

Step 8

The standard quadratic equation using the given set of solutions ${3 - \frac{\sqrt{6}}{3}, 3 + \frac{\sqrt{6}}{3}}$ is $y = x^{2} - 6 x + \frac{25}{3}$ .

$y = x^{2} - 6 x + \frac{25}{3}$

Step 9