Finite Math Examples

$- x^{2} + 3 x + 10 = 0$

Step 1

Subtract $10$ from both sides of the equation.

$- x^{2} + 3 x = - 10$

Step 2

Divide each term in $- x^{2} + 3 x = - 10$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- x^{2} + 3 x = - 10$ by $- 1$ .

$\frac{- x^{2}}{- 1} + \frac{3 x}{- 1} = \frac{- 10}{- 1}$

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

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} + \frac{3 x}{- 1} = \frac{- 10}{- 1}$

Step 2.2.1.2

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

$x^{2} + \frac{3 x}{- 1} = \frac{- 10}{- 1}$

Step 2.2.1.3

Move the negative one from the denominator of $\frac{3 x}{- 1}$ .

$x^{2} - 1 \cdot (3 x) = \frac{- 10}{- 1}$

Step 2.2.1.4

Rewrite $- 1 \cdot (3 x)$ as $- (3 x)$ .

$x^{2} - (3 x) = \frac{- 10}{- 1}$

Step 2.2.1.5

Multiply $3$ by $- 1$ .

$x^{2} - 3 x = \frac{- 10}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $- 10$ by $- 1$ .

$x^{2} - 3 x = 10$

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.

$x^{2} - 3 x + {(- \frac{3}{2})}^{2} = 10 + {(- \frac{3}{2})}^{2}$

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 $- \frac{3}{2}$ .

$x^{2} - 3 x + {(- 1)}^{2} {(\frac{3}{2})}^{2} = 10 + {(- \frac{3}{2})}^{2}$

Step 5.1.1.1.2

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

$x^{2} - 3 x + {(- 1)}^{2} \frac{3^{2}}{2^{2}} = 10 + {(- \frac{3}{2})}^{2}$

Step 5.1.1.2

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

$x^{2} - 3 x + 1 \frac{3^{2}}{2^{2}} = 10 + {(- \frac{3}{2})}^{2}$

Step 5.1.1.3

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

$x^{2} - 3 x + \frac{3^{2}}{2^{2}} = 10 + {(- \frac{3}{2})}^{2}$

Step 5.1.1.4

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

$x^{2} - 3 x + \frac{9}{2^{2}} = 10 + {(- \frac{3}{2})}^{2}$

Step 5.1.1.5

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

$x^{2} - 3 x + \frac{9}{4} = 10 + {(- \frac{3}{2})}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $10 + {(- \frac{3}{2})}^{2}$ .

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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 $- \frac{3}{2}$ .

$x^{2} - 3 x + \frac{9}{4} = 10 + {(- 1)}^{2} {(\frac{3}{2})}^{2}$

Step 5.2.1.1.1.2

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

$x^{2} - 3 x + \frac{9}{4} = 10 + {(- 1)}^{2} \frac{3^{2}}{2^{2}}$

Step 5.2.1.1.2

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

$x^{2} - 3 x + \frac{9}{4} = 10 + 1 \frac{3^{2}}{2^{2}}$

Step 5.2.1.1.3

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

$x^{2} - 3 x + \frac{9}{4} = 10 + \frac{3^{2}}{2^{2}}$

Step 5.2.1.1.4

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

$x^{2} - 3 x + \frac{9}{4} = 10 + \frac{9}{2^{2}}$

Step 5.2.1.1.5

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

$x^{2} - 3 x + \frac{9}{4} = 10 + \frac{9}{4}$

Step 5.2.1.2

To write $10$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} - 3 x + \frac{9}{4} = 10 \cdot \frac{4}{4} + \frac{9}{4}$

Step 5.2.1.3

Combine $10$ and $\frac{4}{4}$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{10 \cdot 4}{4} + \frac{9}{4}$

Step 5.2.1.4

Combine the numerators over the common denominator.

$x^{2} - 3 x + \frac{9}{4} = \frac{10 \cdot 4 + 9}{4}$

Step 5.2.1.5

Simplify the numerator.

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Step 5.2.1.5.1

Multiply $10$ by $4$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{40 + 9}{4}$

Step 5.2.1.5.2

Add $40$ and $9$ .

$x^{2} - 3 x + \frac{9}{4} = \frac{49}{4}$

Step 6

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

${(x - \frac{3}{2})}^{2} = \frac{49}{4}$

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.

$x - \frac{3}{2} = \pm \sqrt{\frac{49}{4}}$

Step 7.2

Simplify $\pm \sqrt{\frac{49}{4}}$ .

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Step 7.2.1

Rewrite $\sqrt{\frac{49}{4}}$ as $\frac{\sqrt{49}}{\sqrt{4}}$ .

$x - \frac{3}{2} = \pm \frac{\sqrt{49}}{\sqrt{4}}$

Step 7.2.2

Simplify the numerator.

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Step 7.2.2.1

Rewrite $49$ as $7^{2}$ .

$x - \frac{3}{2} = \pm \frac{\sqrt{7^{2}}}{\sqrt{4}}$

Step 7.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x - \frac{3}{2} = \pm \frac{7}{\sqrt{4}}$

Step 7.2.3

Simplify the denominator.

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Step 7.2.3.1

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

$x - \frac{3}{2} = \pm \frac{7}{\sqrt{2^{2}}}$

Step 7.2.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x - \frac{3}{2} = \pm \frac{7}{2}$

Step 7.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x - \frac{3}{2} = \frac{7}{2}$

Step 7.3.2

Move all terms not containing $x$ to the right side of the equation.

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Step 7.3.2.1

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

$x = \frac{7}{2} + \frac{3}{2}$

Step 7.3.2.2

Combine the numerators over the common denominator.

$x = \frac{7 + 3}{2}$

Step 7.3.2.3

Add $7$ and $3$ .

$x = \frac{10}{2}$

Step 7.3.2.4

Divide $10$ by $2$ .

$x = 5$

Step 7.3.3

Next, use the negative value of the $\pm$ to find the second solution.

$x - \frac{3}{2} = - \frac{7}{2}$

Step 7.3.4

Move all terms not containing $x$ to the right side of the equation.

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Step 7.3.4.1

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

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

Step 7.3.4.2

Combine the numerators over the common denominator.

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

Step 7.3.4.3

Add $- 7$ and $3$ .

$x = \frac{- 4}{2}$

Step 7.3.4.4

Divide $- 4$ by $2$ .

$x = - 2$

Step 7.3.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = 5, - 2$