Finite Math Examples

$2 x^{2} + 20 x - 30 = 0$

Step 1

Add $30$ to both sides of the equation.

$2 x^{2} + 20 x = 30$

Step 2

Divide each term in $2 x^{2} + 20 x = 30$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 x^{2} + 20 x = 30$ by $2$ .

$\frac{2 x^{2}}{2} + \frac{20 x}{2} = \frac{30}{2}$

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 $2$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} + \frac{20 x}{2} = \frac{30}{2}$

Step 2.2.1.1.2

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

$x^{2} + \frac{20 x}{2} = \frac{30}{2}$

Step 2.2.1.2

Cancel the common factor of $20$ and $2$ .

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Step 2.2.1.2.1

Factor $2$ out of $20 x$ .

$x^{2} + \frac{2 (10 x)}{2} = \frac{30}{2}$

Step 2.2.1.2.2

Cancel the common factors.

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Step 2.2.1.2.2.1

Factor $2$ out of $2$ .

$x^{2} + \frac{2 (10 x)}{2 (1)} = \frac{30}{2}$

Step 2.2.1.2.2.2

Cancel the common factor.

$x^{2} + \frac{2 (10 x)}{2 \cdot 1} = \frac{30}{2}$

Step 2.2.1.2.2.3

Rewrite the expression.

$x^{2} + \frac{10 x}{1} = \frac{30}{2}$

Step 2.2.1.2.2.4

Divide $10 x$ by $1$ .

$x^{2} + 10 x = \frac{30}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $30$ by $2$ .

$x^{2} + 10 x = 15$

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 $b$ .

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

Step 4

Add the term to each side of the equation.

$x^{2} + 10 x + {(5)}^{2} = 15 + {(5)}^{2}$

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

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

$x^{2} + 10 x + 25 = 15 + {(5)}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $15 + {(5)}^{2}$ .

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Step 5.2.1.1

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

$x^{2} + 10 x + 25 = 15 + 25$

Step 5.2.1.2

Add $15$ and $25$ .

$x^{2} + 10 x + 25 = 40$

Step 6

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

${(x + 5)}^{2} = 40$

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.

$x + 5 = \pm \sqrt{40}$

Step 7.2

Simplify $\pm \sqrt{40}$ .

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Step 7.2.1

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 7.2.1.1

Factor $4$ out of $40$ .

$x + 5 = \pm \sqrt{4 (10)}$

Step 7.2.1.2

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

$x + 5 = \pm \sqrt{2^{2} \cdot 10}$

Step 7.2.2

Pull terms out from under the radical.

$x + 5 = \pm 2 \sqrt{10}$

Step 7.3

Subtract $5$ from both sides of the equation.

$x = \pm 2 \sqrt{10} - 5$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm 2 \sqrt{10} - 5$

Decimal Form:

$x = 1.32455532 \dots, - 11.32455532 \dots$