Algebra Examples

$2 - \frac{\sqrt{21}}{3}$ , $2 + \frac{\sqrt{21}}{3}$

Step 1

$x = 2 - \frac{\sqrt{21}}{3}$ and $x = 2 + \frac{\sqrt{21}}{3}$ are the two real distinct solutions for the quadratic equation, which means that $x - 2 - \frac{\sqrt{21}}{3}$ and $x - 2 + \frac{\sqrt{21}}{3}$ are the factors of the quadratic equation.

$(x - 2 - \frac{\sqrt{21}}{3}) (x - 2 + \frac{\sqrt{21}}{3}) = 0$

Step 2

Expand $(x - 2 - \frac{\sqrt{21}}{3}) (x - 2 + \frac{\sqrt{21}}{3})$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x + x \cdot - 2 + x (\frac{\sqrt{21}}{3}) - 2 x - 2 \cdot - 2 - 2 \frac{\sqrt{21}}{3} - \frac{\sqrt{21}}{3} x - \frac{\sqrt{21}}{3} \cdot - 2 - \frac{\sqrt{21}}{3} \cdot \frac{\sqrt{21}}{3} = 0$

Step 3

Simplify terms.

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Step 3.1

Combine the opposite terms in $x \cdot x + x \cdot - 2 + x \frac{\sqrt{21}}{3} - 2 x - 2 \cdot - 2 - 2 \frac{\sqrt{21}}{3} - \frac{\sqrt{21}}{3} x - \frac{\sqrt{21}}{3} \cdot - 2 - \frac{\sqrt{21}}{3} \cdot \frac{\sqrt{21}}{3}$ .

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Step 3.1.1

Reorder the factors in the terms $x \frac{\sqrt{21}}{3}$ and $- \frac{\sqrt{21}}{3} x$ .

$x \cdot x + x \cdot - 2 + \frac{\sqrt{21}}{3} x - 2 x - 2 \cdot - 2 - 2 \frac{\sqrt{21}}{3} - \frac{\sqrt{21}}{3} x - \frac{\sqrt{21}}{3} \cdot - 2 - \frac{\sqrt{21}}{3} \cdot \frac{\sqrt{21}}{3} = 0$

Step 3.1.2

Subtract $\frac{\sqrt{21}}{3} x$ from $\frac{\sqrt{21}}{3} x$ .

$x \cdot x + x \cdot - 2 + 0 - 2 x - 2 \cdot - 2 - 2 \frac{\sqrt{21}}{3} - \frac{\sqrt{21}}{3} \cdot - 2 - \frac{\sqrt{21}}{3} \cdot \frac{\sqrt{21}}{3} = 0$

Step 3.1.3

Add $x \cdot x + x \cdot - 2$ and $0$ .

$x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2 - 2 \frac{\sqrt{21}}{3} - \frac{\sqrt{21}}{3} \cdot - 2 - \frac{\sqrt{21}}{3} \cdot \frac{\sqrt{21}}{3} = 0$

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2 - 2 \frac{\sqrt{21}}{3} - \frac{\sqrt{21}}{3} \cdot - 2 - \frac{\sqrt{21}}{3} \cdot \frac{\sqrt{21}}{3} = 0$

Step 3.2.2

Move $- 2$ to the left of $x$ .

$x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2 - 2 \frac{\sqrt{21}}{3} - \frac{\sqrt{21}}{3} \cdot - 2 - \frac{\sqrt{21}}{3} \cdot \frac{\sqrt{21}}{3} = 0$

Step 3.2.3

Multiply $- 2$ by $- 2$ .

$x^{2} - 2 x - 2 x + 4 - 2 \frac{\sqrt{21}}{3} - \frac{\sqrt{21}}{3} \cdot - 2 - \frac{\sqrt{21}}{3} \cdot \frac{\sqrt{21}}{3} = 0$

Step 3.2.4

Combine $- 2$ and $\frac{\sqrt{21}}{3}$ .

$x^{2} - 2 x - 2 x + 4 + \frac{- 2 \sqrt{21}}{3} - \frac{\sqrt{21}}{3} \cdot - 2 - \frac{\sqrt{21}}{3} \cdot \frac{\sqrt{21}}{3} = 0$

Step 3.2.5

Move the negative in front of the fraction.

$x^{2} - 2 x - 2 x + 4 - \frac{(2) \sqrt{21}}{3} - \frac{\sqrt{21}}{3} \cdot - 2 - \frac{\sqrt{21}}{3} \cdot \frac{\sqrt{21}}{3} = 0$

Step 3.2.6

Multiply $- \frac{\sqrt{21}}{3} \cdot - 2$ .

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Step 3.2.6.1

Multiply $- 2$ by $- 1$ .

$x^{2} - 2 x - 2 x + 4 - \frac{2 \sqrt{21}}{3} + 2 (\frac{\sqrt{21}}{3}) - \frac{\sqrt{21}}{3} \cdot \frac{\sqrt{21}}{3} = 0$

Step 3.2.6.2

Combine $2$ and $\frac{\sqrt{21}}{3}$ .

$x^{2} - 2 x - 2 x + 4 - \frac{2 \sqrt{21}}{3} + \frac{2 \sqrt{21}}{3} - \frac{\sqrt{21}}{3} \cdot \frac{\sqrt{21}}{3} = 0$

Step 3.2.7

Multiply $- \frac{\sqrt{21}}{3} \cdot \frac{\sqrt{21}}{3}$ .

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Step 3.2.7.1

Multiply $\frac{\sqrt{21}}{3}$ by $\frac{\sqrt{21}}{3}$ .

$x^{2} - 2 x - 2 x + 4 - \frac{2 \sqrt{21}}{3} + \frac{2 \sqrt{21}}{3} - \frac{\sqrt{21} \sqrt{21}}{3 \cdot 3} = 0$

Step 3.2.7.2

Raise $\sqrt{21}$ to the power of $1$ .

$x^{2} - 2 x - 2 x + 4 - \frac{2 \sqrt{21}}{3} + \frac{2 \sqrt{21}}{3} - \frac{\sqrt{21} \sqrt{21}}{3 \cdot 3} = 0$

Step 3.2.7.3

Raise $\sqrt{21}$ to the power of $1$ .

$x^{2} - 2 x - 2 x + 4 - \frac{2 \sqrt{21}}{3} + \frac{2 \sqrt{21}}{3} - \frac{\sqrt{21} \sqrt{21}}{3 \cdot 3} = 0$

Step 3.2.7.4

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

$x^{2} - 2 x - 2 x + 4 - \frac{2 \sqrt{21}}{3} + \frac{2 \sqrt{21}}{3} - \frac{{\sqrt{21}}^{1 + 1}}{3 \cdot 3} = 0$

Step 3.2.7.5

Add $1$ and $1$ .

$x^{2} - 2 x - 2 x + 4 - \frac{2 \sqrt{21}}{3} + \frac{2 \sqrt{21}}{3} - \frac{{\sqrt{21}}^{2}}{3 \cdot 3} = 0$

Step 3.2.7.6

Multiply $3$ by $3$ .

$x^{2} - 2 x - 2 x + 4 - \frac{2 \sqrt{21}}{3} + \frac{2 \sqrt{21}}{3} - \frac{{\sqrt{21}}^{2}}{9} = 0$

Step 3.2.8

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 3.2.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$x^{2} - 2 x - 2 x + 4 - \frac{2 \sqrt{21}}{3} + \frac{2 \sqrt{21}}{3} - \frac{{(21^{\frac{1}{2}})}^{2}}{9} = 0$

Step 3.2.8.2

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

$x^{2} - 2 x - 2 x + 4 - \frac{2 \sqrt{21}}{3} + \frac{2 \sqrt{21}}{3} - \frac{21^{\frac{1}{2} \cdot 2}}{9} = 0$

Step 3.2.8.3

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

$x^{2} - 2 x - 2 x + 4 - \frac{2 \sqrt{21}}{3} + \frac{2 \sqrt{21}}{3} - \frac{21^{\frac{2}{2}}}{9} = 0$

Step 3.2.8.4

Cancel the common factor of $2$ .

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Step 3.2.8.4.1

Cancel the common factor.

$x^{2} - 2 x - 2 x + 4 - \frac{2 \sqrt{21}}{3} + \frac{2 \sqrt{21}}{3} - \frac{21^{\frac{2}{2}}}{9} = 0$

Step 3.2.8.4.2

Rewrite the expression.

$x^{2} - 2 x - 2 x + 4 - \frac{2 \sqrt{21}}{3} + \frac{2 \sqrt{21}}{3} - \frac{21}{9} = 0$

Step 3.2.8.5

Evaluate the exponent.

$x^{2} - 2 x - 2 x + 4 - \frac{2 \sqrt{21}}{3} + \frac{2 \sqrt{21}}{3} - \frac{21}{9} = 0$

Step 3.2.9

Cancel the common factor of $21$ and $9$ .

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Step 3.2.9.1

Factor $3$ out of $21$ .

$x^{2} - 2 x - 2 x + 4 - \frac{2 \sqrt{21}}{3} + \frac{2 \sqrt{21}}{3} - \frac{3 (7)}{9} = 0$

Step 3.2.9.2

Cancel the common factors.

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Step 3.2.9.2.1

Factor $3$ out of $9$ .

$x^{2} - 2 x - 2 x + 4 - \frac{2 \sqrt{21}}{3} + \frac{2 \sqrt{21}}{3} - \frac{3 \cdot 7}{3 \cdot 3} = 0$

Step 3.2.9.2.2

Cancel the common factor.

$x^{2} - 2 x - 2 x + 4 - \frac{2 \sqrt{21}}{3} + \frac{2 \sqrt{21}}{3} - \frac{3 \cdot 7}{3 \cdot 3} = 0$

Step 3.2.9.2.3

Rewrite the expression.

$x^{2} - 2 x - 2 x + 4 - \frac{2 \sqrt{21}}{3} + \frac{2 \sqrt{21}}{3} - \frac{7}{3} = 0$

Step 3.3

Simplify by adding terms.

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Step 3.3.1

Combine the opposite terms in $x^{2} - 2 x - 2 x + 4 - \frac{2 \sqrt{21}}{3} + \frac{2 \sqrt{21}}{3} - \frac{7}{3}$ .

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Step 3.3.1.1

Add $- \frac{2 \sqrt{21}}{3}$ and $\frac{2 \sqrt{21}}{3}$ .

$x^{2} - 2 x - 2 x + 4 + 0 - \frac{7}{3} = 0$

Step 3.3.1.2

Add $x^{2} - 2 x - 2 x + 4$ and $0$ .

$x^{2} - 2 x - 2 x + 4 - \frac{7}{3} = 0$

Step 3.3.2

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

$x^{2} - 4 x + 4 - \frac{7}{3} = 0$

Step 4

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{2} - 4 x + 4 \cdot \frac{3}{3} - \frac{7}{3} = 0$

Step 5

Combine $4$ and $\frac{3}{3}$ .

$x^{2} - 4 x + \frac{4 \cdot 3}{3} - \frac{7}{3} = 0$

Step 6

Combine the numerators over the common denominator.

$x^{2} - 4 x + \frac{4 \cdot 3 - 7}{3} = 0$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $4$ by $3$ .

$x^{2} - 4 x + \frac{12 - 7}{3} = 0$

Step 7.2

Subtract $7$ from $12$ .

$x^{2} - 4 x + \frac{5}{3} = 0$

Step 8

The standard quadratic equation using the given set of solutions ${2 - \frac{\sqrt{21}}{3}, 2 + \frac{\sqrt{21}}{3}}$ is $y = x^{2} - 4 x + \frac{5}{3}$ .

$y = x^{2} - 4 x + \frac{5}{3}$

Step 9