Finite Math Examples

$3 x^{2} - 9 x - 50 = 0$

Step 1

Add $50$ to both sides of the equation.

$3 x^{2} - 9 x = 50$

Step 2

Divide each term in $3 x^{2} - 9 x = 50$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 x^{2} - 9 x = 50$ by $3$ .

$\frac{3 x^{2}}{3} + \frac{- 9 x}{3} = \frac{50}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $3$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} + \frac{- 9 x}{3} = \frac{50}{3}$

Step 2.2.1.1.2

Divide $x^{2}$ by $1$ .

$x^{2} + \frac{- 9 x}{3} = \frac{50}{3}$

Step 2.2.1.2

Cancel the common factor of $- 9$ and $3$ .

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Step 2.2.1.2.1

Factor $3$ out of $- 9 x$ .

$x^{2} + \frac{3 (- 3 x)}{3} = \frac{50}{3}$

Step 2.2.1.2.2

Cancel the common factors.

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Step 2.2.1.2.2.1

Factor $3$ out of $3$ .

$x^{2} + \frac{3 (- 3 x)}{3 (1)} = \frac{50}{3}$

Step 2.2.1.2.2.2

Cancel the common factor.

$x^{2} + \frac{3 (- 3 x)}{3 \cdot 1} = \frac{50}{3}$

Step 2.2.1.2.2.3

Rewrite the expression.

$x^{2} + \frac{- 3 x}{1} = \frac{50}{3}$

Step 2.2.1.2.2.4

Divide $- 3 x$ by $1$ .

$x^{2} - 3 x = \frac{50}{3}$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{3}{2})}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} - 3 x + {(- \frac{3}{2})}^{2} = \frac{50}{3} + {(- \frac{3}{2})}^{2}$

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1.1.1.1

Apply the product rule to $- \frac{3}{2}$ .

$x^{2} - 3 x + {(- 1)}^{2} {(\frac{3}{2})}^{2} = \frac{50}{3} + {(- \frac{3}{2})}^{2}$

Step 5.1.1.1.2

Apply the product rule to $\frac{3}{2}$ .

$x^{2} - 3 x + {(- 1)}^{2} \frac{3^{2}}{2^{2}} = \frac{50}{3} + {(- \frac{3}{2})}^{2}$

Step 5.1.1.2

Raise $- 1$ to the power of $2$ .

$x^{2} - 3 x + 1 \frac{3^{2}}{2^{2}} = \frac{50}{3} + {(- \frac{3}{2})}^{2}$

Step 5.1.1.3

Multiply $\frac{3^{2}}{2^{2}}$ by $1$ .

$x^{2} - 3 x + \frac{3^{2}}{2^{2}} = \frac{50}{3} + {(- \frac{3}{2})}^{2}$

Step 5.1.1.4

Raise $3$ to the power of $2$ .

$x^{2} - 3 x + \frac{9}{2^{2}} = \frac{50}{3} + {(- \frac{3}{2})}^{2}$

Step 5.1.1.5

Raise $2$ to the power of $2$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{50}{3} + {(- \frac{3}{2})}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{50}{3} + {(- \frac{3}{2})}^{2}$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.1.1.1.1

Apply the product rule to $- \frac{3}{2}$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{50}{3} + {(- 1)}^{2} {(\frac{3}{2})}^{2}$

Step 5.2.1.1.1.2

Apply the product rule to $\frac{3}{2}$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{50}{3} + {(- 1)}^{2} \frac{3^{2}}{2^{2}}$

Step 5.2.1.1.2

Raise $- 1$ to the power of $2$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{50}{3} + 1 \frac{3^{2}}{2^{2}}$

Step 5.2.1.1.3

Multiply $\frac{3^{2}}{2^{2}}$ by $1$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{50}{3} + \frac{3^{2}}{2^{2}}$

Step 5.2.1.1.4

Raise $3$ to the power of $2$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{50}{3} + \frac{9}{2^{2}}$

Step 5.2.1.1.5

Raise $2$ to the power of $2$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{50}{3} + \frac{9}{4}$

Step 5.2.1.2

To write $\frac{50}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{50}{3} \cdot \frac{4}{4} + \frac{9}{4}$

Step 5.2.1.3

To write $\frac{9}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{50}{3} \cdot \frac{4}{4} + \frac{9}{4} \cdot \frac{3}{3}$

Step 5.2.1.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.1.4.1

Multiply $\frac{50}{3}$ by $\frac{4}{4}$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{50 \cdot 4}{3 \cdot 4} + \frac{9}{4} \cdot \frac{3}{3}$

Step 5.2.1.4.2

Multiply $3$ by $4$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{50 \cdot 4}{12} + \frac{9}{4} \cdot \frac{3}{3}$

Step 5.2.1.4.3

Multiply $\frac{9}{4}$ by $\frac{3}{3}$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{50 \cdot 4}{12} + \frac{9 \cdot 3}{4 \cdot 3}$

Step 5.2.1.4.4

Multiply $4$ by $3$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{50 \cdot 4}{12} + \frac{9 \cdot 3}{12}$

Step 5.2.1.5

Combine the numerators over the common denominator.

$x^{2} - 3 x + \frac{9}{4} = \frac{50 \cdot 4 + 9 \cdot 3}{12}$

Step 5.2.1.6

Simplify the numerator.

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Step 5.2.1.6.1

Multiply $50$ by $4$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{200 + 9 \cdot 3}{12}$

Step 5.2.1.6.2

Multiply $9$ by $3$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{200 + 27}{12}$

Step 5.2.1.6.3

Add $200$ and $27$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{227}{12}$

Step 6

Factor the perfect trinomial square into ${(x - \frac{3}{2})}^{2}$ .

${(x - \frac{3}{2})}^{2} = \frac{227}{12}$

Step 7

Solve the equation for $x$ .

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Step 7.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x - \frac{3}{2} = \pm \sqrt{\frac{227}{12}}$

Step 7.2

Simplify $\pm \sqrt{\frac{227}{12}}$ .

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Step 7.2.1

Rewrite $\sqrt{\frac{227}{12}}$ as $\frac{\sqrt{227}}{\sqrt{12}}$ .

$x - \frac{3}{2} = \pm \frac{\sqrt{227}}{\sqrt{12}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 7.2.2.1.1

Factor $4$ out of $12$ .

$x - \frac{3}{2} = \pm \frac{\sqrt{227}}{\sqrt{4 (3)}}$

Step 7.2.2.1.2

Rewrite $4$ as $2^{2}$ .

$x - \frac{3}{2} = \pm \frac{\sqrt{227}}{\sqrt{2^{2} \cdot 3}}$

Step 7.2.2.2

Pull terms out from under the radical.

$x - \frac{3}{2} = \pm \frac{\sqrt{227}}{2 \sqrt{3}}$

Step 7.2.3

Multiply $\frac{\sqrt{227}}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x - \frac{3}{2} = \pm \frac{\sqrt{227}}{2 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 7.2.4

Combine and simplify the denominator.

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Step 7.2.4.1

Multiply $\frac{\sqrt{227}}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x - \frac{3}{2} = \pm \frac{\sqrt{227} \sqrt{3}}{2 \sqrt{3} \sqrt{3}}$

Step 7.2.4.2

Move $\sqrt{3}$ .

$x - \frac{3}{2} = \pm \frac{\sqrt{227} \sqrt{3}}{2 (\sqrt{3} \sqrt{3})}$

Step 7.2.4.3

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

$x - \frac{3}{2} = \pm \frac{\sqrt{227} \sqrt{3}}{2 ({\sqrt{3}}^{1} \sqrt{3})}$

Step 7.2.4.4

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

$x - \frac{3}{2} = \pm \frac{\sqrt{227} \sqrt{3}}{2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

Step 7.2.4.5

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

$x - \frac{3}{2} = \pm \frac{\sqrt{227} \sqrt{3}}{2 {\sqrt{3}}^{1 + 1}}$

Step 7.2.4.6

Add $1$ and $1$ .

$x - \frac{3}{2} = \pm \frac{\sqrt{227} \sqrt{3}}{2 {\sqrt{3}}^{2}}$

Step 7.2.4.7

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

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Step 7.2.4.7.1

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

$x - \frac{3}{2} = \pm \frac{\sqrt{227} \sqrt{3}}{2 {(3^{\frac{1}{2}})}^{2}}$

Step 7.2.4.7.2

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

$x - \frac{3}{2} = \pm \frac{\sqrt{227} \sqrt{3}}{2 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 7.2.4.7.3

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

$x - \frac{3}{2} = \pm \frac{\sqrt{227} \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

Step 7.2.4.7.4

Cancel the common factor of $2$ .

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Step 7.2.4.7.4.1

Cancel the common factor.

$x - \frac{3}{2} = \pm \frac{\sqrt{227} \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

Step 7.2.4.7.4.2

Rewrite the expression.

$x - \frac{3}{2} = \pm \frac{\sqrt{227} \sqrt{3}}{2 \cdot 3^{1}}$

Step 7.2.4.7.5

Evaluate the exponent.

$x - \frac{3}{2} = \pm \frac{\sqrt{227} \sqrt{3}}{2 \cdot 3}$

Step 7.2.5

Simplify the numerator.

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Step 7.2.5.1

Combine using the product rule for radicals.

$x - \frac{3}{2} = \pm \frac{\sqrt{227 \cdot 3}}{2 \cdot 3}$

Step 7.2.5.2

Multiply $227$ by $3$ .

$x - \frac{3}{2} = \pm \frac{\sqrt{681}}{2 \cdot 3}$

Step 7.2.6

Multiply $2$ by $3$ .

$x - \frac{3}{2} = \pm \frac{\sqrt{681}}{6}$

Step 7.3

Add $\frac{3}{2}$ to both sides of the equation.

$x = \pm \frac{\sqrt{681}}{6} + \frac{3}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{681}}{6} + \frac{3}{2}$

Decimal Form:

$x = 5.84932945 \dots, - 2.84932945 \dots$