Finite Math Examples

$0.35 (x^{2}) - 32.9 x + 975 = 45$

Step 1

Subtract $45$ from both sides of the equation.

$0.35 (x^{2}) - 32.9 x + 975 - 45 = 0$

Step 2

Subtract $45$ from $975$ .

$0.35 x^{2} - 32.9 x + 930 = 0$

Step 3

Factor $0.05$ out of $0.35 x^{2} - 32.9 x + 930$ .

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Step 3.1

Factor $0.05$ out of $0.35 x^{2}$ .

$0.05 (7 x^{2}) - 32.9 x + 930 = 0$

Step 3.2

Factor $0.05$ out of $- 32.9 x$ .

$0.05 (7 x^{2}) + 0.05 (- 658 x) + 930 = 0$

Step 3.3

Factor $0.05$ out of $930$ .

$0.05 (7 x^{2}) + 0.05 (- 658 x) + 0.05 (18600) = 0$

Step 3.4

Factor $0.05$ out of $0.05 (7 x^{2}) + 0.05 (- 658 x)$ .

$0.05 (7 x^{2} - 658 x) + 0.05 (18600) = 0$

Step 3.5

Factor $0.05$ out of $0.05 (7 x^{2} - 658 x) + 0.05 (18600)$ .

$0.05 (7 x^{2} - 658 x + 18600) = 0$

Step 4

Divide each term in $0.05 (7 x^{2} - 658 x + 18600) = 0$ by $0.05$ and simplify.

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Step 4.1

Divide each term in $0.05 (7 x^{2} - 658 x + 18600) = 0$ by $0.05$ .

$\frac{0.05 (7 x^{2} - 658 x + 18600)}{0.05} = \frac{0}{0.05}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $0.05$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{0.05 (7 x^{2} - 658 x + 18600)}{0.05} = \frac{0}{0.05}$

Step 4.2.1.2

Divide $7 x^{2} - 658 x + 18600$ by $1$ .

$7 x^{2} - 658 x + 18600 = \frac{0}{0.05}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $0.05$ .

$7 x^{2} - 658 x + 18600 = 0$

Step 5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6

Substitute the values $a = 7$ , $b = - 658$ , and $c = 18600$ into the quadratic formula and solve for $x$ .

$\frac{658 \pm \sqrt{{(- 658)}^{2} - 4 \cdot (7 \cdot 18600)}}{2 \cdot 7}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 658$ to the power of $2$ .

$x = \frac{658 \pm \sqrt{432964 - 4 \cdot 7 \cdot 18600}}{2 \cdot 7}$

Step 7.1.2

Multiply $- 4 \cdot 7 \cdot 18600$ .

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Step 7.1.2.1

Multiply $- 4$ by $7$ .

$x = \frac{658 \pm \sqrt{432964 - 28 \cdot 18600}}{2 \cdot 7}$

Step 7.1.2.2

Multiply $- 28$ by $18600$ .

$x = \frac{658 \pm \sqrt{432964 - 520800}}{2 \cdot 7}$

Step 7.1.3

Subtract $520800$ from $432964$ .

$x = \frac{658 \pm \sqrt{- 87836}}{2 \cdot 7}$

Step 7.1.4

Rewrite $- 87836$ as $- 1 (87836)$ .

$x = \frac{658 \pm \sqrt{- 1 \cdot 87836}}{2 \cdot 7}$

Step 7.1.5

Rewrite $\sqrt{- 1 (87836)}$ as $\sqrt{- 1} \cdot \sqrt{87836}$ .

$x = \frac{658 \pm \sqrt{- 1} \cdot \sqrt{87836}}{2 \cdot 7}$

Step 7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{658 \pm i \cdot \sqrt{87836}}{2 \cdot 7}$

Step 7.1.7

Rewrite $87836$ as $2^{2} \cdot 21959$ .

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Step 7.1.7.1

Factor $4$ out of $87836$ .

$x = \frac{658 \pm i \cdot \sqrt{4 (21959)}}{2 \cdot 7}$

Step 7.1.7.2

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

$x = \frac{658 \pm i \cdot \sqrt{2^{2} \cdot 21959}}{2 \cdot 7}$

Step 7.1.8

Pull terms out from under the radical.

$x = \frac{658 \pm i \cdot (2 \sqrt{21959})}{2 \cdot 7}$

Step 7.1.9

Move $2$ to the left of $i$ .

$x = \frac{658 \pm 2 i \sqrt{21959}}{2 \cdot 7}$

Step 7.2

Multiply $2$ by $7$ .

$x = \frac{658 \pm 2 i \sqrt{21959}}{14}$

Step 7.3

Simplify $\frac{658 \pm 2 i \sqrt{21959}}{14}$ .

$x = \frac{329 \pm i \sqrt{21959}}{7}$

Step 8

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 658$ to the power of $2$ .

$x = \frac{658 \pm \sqrt{432964 - 4 \cdot 7 \cdot 18600}}{2 \cdot 7}$

Step 8.1.2

Multiply $- 4 \cdot 7 \cdot 18600$ .

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Step 8.1.2.1

Multiply $- 4$ by $7$ .

$x = \frac{658 \pm \sqrt{432964 - 28 \cdot 18600}}{2 \cdot 7}$

Step 8.1.2.2

Multiply $- 28$ by $18600$ .

$x = \frac{658 \pm \sqrt{432964 - 520800}}{2 \cdot 7}$

Step 8.1.3

Subtract $520800$ from $432964$ .

$x = \frac{658 \pm \sqrt{- 87836}}{2 \cdot 7}$

Step 8.1.4

Rewrite $- 87836$ as $- 1 (87836)$ .

$x = \frac{658 \pm \sqrt{- 1 \cdot 87836}}{2 \cdot 7}$

Step 8.1.5

Rewrite $\sqrt{- 1 (87836)}$ as $\sqrt{- 1} \cdot \sqrt{87836}$ .

$x = \frac{658 \pm \sqrt{- 1} \cdot \sqrt{87836}}{2 \cdot 7}$

Step 8.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{658 \pm i \cdot \sqrt{87836}}{2 \cdot 7}$

Step 8.1.7

Rewrite $87836$ as $2^{2} \cdot 21959$ .

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Step 8.1.7.1

Factor $4$ out of $87836$ .

$x = \frac{658 \pm i \cdot \sqrt{4 (21959)}}{2 \cdot 7}$

Step 8.1.7.2

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

$x = \frac{658 \pm i \cdot \sqrt{2^{2} \cdot 21959}}{2 \cdot 7}$

Step 8.1.8

Pull terms out from under the radical.

$x = \frac{658 \pm i \cdot (2 \sqrt{21959})}{2 \cdot 7}$

Step 8.1.9

Move $2$ to the left of $i$ .

$x = \frac{658 \pm 2 i \sqrt{21959}}{2 \cdot 7}$

Step 8.2

Multiply $2$ by $7$ .

$x = \frac{658 \pm 2 i \sqrt{21959}}{14}$

Step 8.3

Simplify $\frac{658 \pm 2 i \sqrt{21959}}{14}$ .

$x = \frac{329 \pm i \sqrt{21959}}{7}$

Step 8.4

Change the $\pm$ to $+$ .

$x = \frac{329 + i \sqrt{21959}}{7}$

Step 9

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $- 658$ to the power of $2$ .

$x = \frac{658 \pm \sqrt{432964 - 4 \cdot 7 \cdot 18600}}{2 \cdot 7}$

Step 9.1.2

Multiply $- 4 \cdot 7 \cdot 18600$ .

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Step 9.1.2.1

Multiply $- 4$ by $7$ .

$x = \frac{658 \pm \sqrt{432964 - 28 \cdot 18600}}{2 \cdot 7}$

Step 9.1.2.2

Multiply $- 28$ by $18600$ .

$x = \frac{658 \pm \sqrt{432964 - 520800}}{2 \cdot 7}$

Step 9.1.3

Subtract $520800$ from $432964$ .

$x = \frac{658 \pm \sqrt{- 87836}}{2 \cdot 7}$

Step 9.1.4

Rewrite $- 87836$ as $- 1 (87836)$ .

$x = \frac{658 \pm \sqrt{- 1 \cdot 87836}}{2 \cdot 7}$

Step 9.1.5

Rewrite $\sqrt{- 1 (87836)}$ as $\sqrt{- 1} \cdot \sqrt{87836}$ .

$x = \frac{658 \pm \sqrt{- 1} \cdot \sqrt{87836}}{2 \cdot 7}$

Step 9.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{658 \pm i \cdot \sqrt{87836}}{2 \cdot 7}$

Step 9.1.7

Rewrite $87836$ as $2^{2} \cdot 21959$ .

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Step 9.1.7.1

Factor $4$ out of $87836$ .

$x = \frac{658 \pm i \cdot \sqrt{4 (21959)}}{2 \cdot 7}$

Step 9.1.7.2

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

$x = \frac{658 \pm i \cdot \sqrt{2^{2} \cdot 21959}}{2 \cdot 7}$

Step 9.1.8

Pull terms out from under the radical.

$x = \frac{658 \pm i \cdot (2 \sqrt{21959})}{2 \cdot 7}$

Step 9.1.9

Move $2$ to the left of $i$ .

$x = \frac{658 \pm 2 i \sqrt{21959}}{2 \cdot 7}$

Step 9.2

Multiply $2$ by $7$ .

$x = \frac{658 \pm 2 i \sqrt{21959}}{14}$

Step 9.3

Simplify $\frac{658 \pm 2 i \sqrt{21959}}{14}$ .

$x = \frac{329 \pm i \sqrt{21959}}{7}$

Step 9.4

Change the $\pm$ to $-$ .

$x = \frac{329 - i \sqrt{21959}}{7}$

Step 10

The final answer is the combination of both solutions.

$x = \frac{329 + i \sqrt{21959}}{7}, \frac{329 - i \sqrt{21959}}{7}$