Algebra Examples

$- \sqrt{3}$ , $8 \sqrt{3}$

Step 1

$x = - \sqrt{3}$ and $x = 8 \sqrt{3}$ are the two real distinct solutions for the quadratic equation, which means that $x + \sqrt{3}$ and $x - 8 \sqrt{3}$ are the factors of the quadratic equation.

$(x + \sqrt{3}) (x - 8 \sqrt{3}) = 0$

Step 2

Expand $(x + \sqrt{3}) (x - 8 \sqrt{3})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$x (x - 8 \sqrt{3}) + \sqrt{3} (x - 8 \sqrt{3}) = 0$

Step 2.2

Apply the distributive property.

$x \cdot x + x (- 8 \sqrt{3}) + \sqrt{3} (x - 8 \sqrt{3}) = 0$

Step 2.3

Apply the distributive property.

$x \cdot x + x (- 8 \sqrt{3}) + \sqrt{3} x + \sqrt{3} (- 8 \sqrt{3}) = 0$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

$x^{2} + x (- 8 \sqrt{3}) + \sqrt{3} x + \sqrt{3} (- 8 \sqrt{3}) = 0$

Step 3.1.2

Multiply $\sqrt{3} (- 8 \sqrt{3})$ .

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Step 3.1.2.1

Raise $\sqrt{3}$ to the power of $1$ .

$x^{2} + x (- 8 \sqrt{3}) + \sqrt{3} x - 8 (\sqrt{3} \sqrt{3}) = 0$

Step 3.1.2.2

Raise $\sqrt{3}$ to the power of $1$ .

$x^{2} + x (- 8 \sqrt{3}) + \sqrt{3} x - 8 (\sqrt{3} \sqrt{3}) = 0$

Step 3.1.2.3

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

$x^{2} + x (- 8 \sqrt{3}) + \sqrt{3} x - 8 {\sqrt{3}}^{1 + 1} = 0$

Step 3.1.2.4

Add $1$ and $1$ .

$x^{2} + x (- 8 \sqrt{3}) + \sqrt{3} x - 8 {\sqrt{3}}^{2} = 0$

Step 3.1.3

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x^{2} + x (- 8 \sqrt{3}) + \sqrt{3} x - 8 {(3^{\frac{1}{2}})}^{2} = 0$

Step 3.1.3.2

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

$x^{2} + x (- 8 \sqrt{3}) + \sqrt{3} x - 8 \cdot 3^{\frac{1}{2} \cdot 2} = 0$

Step 3.1.3.3

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

$x^{2} + x (- 8 \sqrt{3}) + \sqrt{3} x - 8 \cdot 3^{\frac{2}{2}} = 0$

Step 3.1.3.4

Cancel the common factor of $2$ .

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Step 3.1.3.4.1

Cancel the common factor.

$x^{2} + x (- 8 \sqrt{3}) + \sqrt{3} x - 8 \cdot 3^{\frac{2}{2}} = 0$

Step 3.1.3.4.2

Rewrite the expression.

$x^{2} + x (- 8 \sqrt{3}) + \sqrt{3} x - 8 \cdot 3 = 0$

Step 3.1.3.5

Evaluate the exponent.

$x^{2} + x (- 8 \sqrt{3}) + \sqrt{3} x - 8 \cdot 3 = 0$

Step 3.1.4

Multiply $- 8$ by $3$ .

$x^{2} + x (- 8 \sqrt{3}) + \sqrt{3} x - 24 = 0$

Step 3.2

Reorder the factors of $\sqrt{3} x$ .

$x^{2} + x (- 8 \sqrt{3}) + x \sqrt{3} - 24 = 0$

Step 3.3

Add $x (- 8 \sqrt{3})$ and $x \sqrt{3}$ .

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Step 3.3.1

Reorder $x$ and $- 8$ .

$x^{2} - 8 \cdot (x \sqrt{3}) + x \sqrt{3} - 24 = 0$

Step 3.3.2

Add $- 8 \cdot x \sqrt{3}$ and $x \sqrt{3}$ .

$x^{2} - 7 x \sqrt{3} - 24 = 0$

Step 4

The standard quadratic equation using the given set of solutions ${- \sqrt{3}, 8 \sqrt{3}}$ is $y = x^{2} - 7 x \sqrt{3} - 24$ .

$y = x^{2} - 7 x \sqrt{3} - 24$

Step 5