Finite Math Examples

$5 x^{2} + 4 x = 1$

Step 1

Divide each term in $5 x^{2} + 4 x = 1$ by $5$ and simplify.

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Step 1.1

Divide each term in $5 x^{2} + 4 x = 1$ by $5$ .

$\frac{5 x^{2}}{5} + \frac{4 x}{5} = \frac{1}{5}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $5$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{5 x^{2}}{5} + \frac{4 x}{5} = \frac{1}{5}$

Step 1.2.1.2

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

$x^{2} + \frac{4 x}{5} = \frac{1}{5}$

Step 2

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

${(\frac{b}{2})}^{2} = {(\frac{2}{5})}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} + \frac{4 x}{5} + {(\frac{2}{5})}^{2} = \frac{1}{5} + {(\frac{2}{5})}^{2}$

Step 4

Simplify the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Apply the product rule to $\frac{2}{5}$ .

$x^{2} + \frac{4 x}{5} + \frac{2^{2}}{5^{2}} = \frac{1}{5} + {(\frac{2}{5})}^{2}$

Step 4.1.1.2

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

$x^{2} + \frac{4 x}{5} + \frac{4}{5^{2}} = \frac{1}{5} + {(\frac{2}{5})}^{2}$

Step 4.1.1.3

Raise $5$ to the power of $2$ .

$x^{2} + \frac{4 x}{5} + \frac{4}{25} = \frac{1}{5} + {(\frac{2}{5})}^{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{1}{5} + {(\frac{2}{5})}^{2}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the product rule to $\frac{2}{5}$ .

$x^{2} + \frac{4 x}{5} + \frac{4}{25} = \frac{1}{5} + \frac{2^{2}}{5^{2}}$

Step 4.2.1.1.2

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

$x^{2} + \frac{4 x}{5} + \frac{4}{25} = \frac{1}{5} + \frac{4}{5^{2}}$

Step 4.2.1.1.3

Raise $5$ to the power of $2$ .

$x^{2} + \frac{4 x}{5} + \frac{4}{25} = \frac{1}{5} + \frac{4}{25}$

Step 4.2.1.2

To write $\frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x^{2} + \frac{4 x}{5} + \frac{4}{25} = \frac{1}{5} \cdot \frac{5}{5} + \frac{4}{25}$

Step 4.2.1.3

Write each expression with a common denominator of $25$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.1.3.1

Multiply $\frac{1}{5}$ by $\frac{5}{5}$ .

$x^{2} + \frac{4 x}{5} + \frac{4}{25} = \frac{5}{5 \cdot 5} + \frac{4}{25}$

Step 4.2.1.3.2

Multiply $5$ by $5$ .

$x^{2} + \frac{4 x}{5} + \frac{4}{25} = \frac{5}{25} + \frac{4}{25}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$x^{2} + \frac{4 x}{5} + \frac{4}{25} = \frac{5 + 4}{25}$

Step 4.2.1.5

Add $5$ and $4$ .

$x^{2} + \frac{4 x}{5} + \frac{4}{25} = \frac{9}{25}$

Step 5

Factor the perfect trinomial square into ${(x + \frac{2}{5})}^{2}$ .

${(x + \frac{2}{5})}^{2} = \frac{9}{25}$

Step 6

Solve the equation for $x$ .

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Step 6.1

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

$x + \frac{2}{5} = \pm \sqrt{\frac{9}{25}}$

Step 6.2

Simplify $\pm \sqrt{\frac{9}{25}}$ .

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Step 6.2.1

Rewrite $\sqrt{\frac{9}{25}}$ as $\frac{\sqrt{9}}{\sqrt{25}}$ .

$x + \frac{2}{5} = \pm \frac{\sqrt{9}}{\sqrt{25}}$

Step 6.2.2

Simplify the numerator.

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Step 6.2.2.1

Rewrite $9$ as $3^{2}$ .

$x + \frac{2}{5} = \pm \frac{\sqrt{3^{2}}}{\sqrt{25}}$

Step 6.2.2.2

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

$x + \frac{2}{5} = \pm \frac{3}{\sqrt{25}}$

Step 6.2.3

Simplify the denominator.

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Step 6.2.3.1

Rewrite $25$ as $5^{2}$ .

$x + \frac{2}{5} = \pm \frac{3}{\sqrt{5^{2}}}$

Step 6.2.3.2

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

$x + \frac{2}{5} = \pm \frac{3}{5}$

Step 6.3

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

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Step 6.3.1

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

$x + \frac{2}{5} = \frac{3}{5}$

Step 6.3.2

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

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Step 6.3.2.1

Subtract $\frac{2}{5}$ from both sides of the equation.

$x = \frac{3}{5} - \frac{2}{5}$

Step 6.3.2.2

Combine the numerators over the common denominator.

$x = \frac{3 - 2}{5}$

Step 6.3.2.3

Subtract $2$ from $3$ .

$x = \frac{1}{5}$

Step 6.3.3

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

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

Step 6.3.4

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

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Step 6.3.4.1

Subtract $\frac{2}{5}$ from both sides of the equation.

$x = - \frac{3}{5} - \frac{2}{5}$

Step 6.3.4.2

Combine the numerators over the common denominator.

$x = \frac{- 3 - 2}{5}$

Step 6.3.4.3

Subtract $2$ from $- 3$ .

$x = \frac{- 5}{5}$

Step 6.3.4.4

Divide $- 5$ by $5$ .

$x = - 1$

Step 6.3.5

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

$x = \frac{1}{5}, - 1$