Finite Math Examples

$i + 24 i x + 2 i x^{2}$

Step 1

Set $i + 24 i x + 2 i x^{2}$ equal to $0$ .

$i + 24 i x + 2 i x^{2} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Use the quadratic formula to find the solutions.

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

Step 2.2

Substitute the values $a = 2 i$ , $b = 24 i$ , and $c = i$ into the quadratic formula and solve for $x$ .

$\frac{- 24 i \pm \sqrt{{(24 i)}^{2} - 4 \cdot (2 i \cdot i)}}{2 (2 i)}$

Step 2.3

Simplify.

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Step 2.3.1

Multiply the numerator and denominator of $\frac{- 24 i \pm \sqrt{{(24 i)}^{2} - 4 \cdot (2 i \cdot i)}}{4 i}$ by the conjugate of $4 i$ to make the denominator real.

$x = \frac{- 24 i \pm \sqrt{{(24 i)}^{2} - 4 \cdot (2 i \cdot i)}}{4 i} \cdot \frac{i}{i}$

Step 2.3.2

Multiply.

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Step 2.3.2.1

Combine.

$x = \frac{(- 24 i \pm \sqrt{{(24 i)}^{2} - 4 \cdot (2 i \cdot i)}) i}{4 i i}$

Step 2.3.2.2

Simplify the numerator.

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Step 2.3.2.2.1

Apply the product rule to $24 i$ .

$x = \frac{(- 24 i \pm \sqrt{24^{2} i^{2} - 4 \cdot (2 i \cdot i)}) i}{4 i i}$

Step 2.3.2.2.2

Raise $24$ to the power of $2$ .

$x = \frac{(- 24 i \pm \sqrt{576 i^{2} - 4 \cdot (2 i \cdot i)}) i}{4 i i}$

Step 2.3.2.2.3

Rewrite $i^{2}$ as $- 1$ .

$x = \frac{(- 24 i \pm \sqrt{576 \cdot - 1 - 4 \cdot (2 i \cdot i)}) i}{4 i i}$

Step 2.3.2.2.4

Multiply $576$ by $- 1$ .

$x = \frac{(- 24 i \pm \sqrt{- 576 - 4 \cdot (2 i \cdot i)}) i}{4 i i}$

Step 2.3.2.2.5

Multiply $2 i i$ .

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Step 2.3.2.2.5.1

Raise $i$ to the power of $1$ .

$x = \frac{(- 24 i \pm \sqrt{- 576 - 4 \cdot (2 (i i))}) i}{4 i i}$

Step 2.3.2.2.5.2

Raise $i$ to the power of $1$ .

$x = \frac{(- 24 i \pm \sqrt{- 576 - 4 \cdot (2 (i i))}) i}{4 i i}$

Step 2.3.2.2.5.3

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

$x = \frac{(- 24 i \pm \sqrt{- 576 - 4 \cdot (2 i^{1 + 1})}) i}{4 i i}$

Step 2.3.2.2.5.4

Add $1$ and $1$ .

$x = \frac{(- 24 i \pm \sqrt{- 576 - 4 \cdot (2 i^{2})}) i}{4 i i}$

Step 2.3.2.2.6

Rewrite $i^{2}$ as $- 1$ .

$x = \frac{(- 24 i \pm \sqrt{- 576 - 4 \cdot (2 \cdot - 1)}) i}{4 i i}$

Step 2.3.2.2.7

Multiply $- 4 (2 \cdot - 1)$ .

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Step 2.3.2.2.7.1

Multiply $2$ by $- 1$ .

$x = \frac{(- 24 i \pm \sqrt{- 576 - 4 \cdot - 2}) i}{4 i i}$

Step 2.3.2.2.7.2

Multiply $- 4$ by $- 2$ .

$x = \frac{(- 24 i \pm \sqrt{- 576 + 8}) i}{4 i i}$

Step 2.3.2.2.8

Add $- 576$ and $8$ .

$x = \frac{(- 24 i \pm \sqrt{- 568}) i}{4 i i}$

Step 2.3.2.2.9

Rewrite $- 568$ as $- 1 (568)$ .

$x = \frac{(- 24 i \pm \sqrt{- 1 \cdot 568}) i}{4 i i}$

Step 2.3.2.2.10

Rewrite $\sqrt{- 1 (568)}$ as $\sqrt{- 1} \cdot \sqrt{568}$ .

$x = \frac{(- 24 i \pm \sqrt{- 1} \cdot \sqrt{568}) i}{4 i i}$

Step 2.3.2.2.11

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

$x = \frac{(- 24 i \pm i \cdot \sqrt{568}) i}{4 i i}$

Step 2.3.2.2.12

Rewrite $568$ as $2^{2} \cdot 142$ .

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Step 2.3.2.2.12.1

Factor $4$ out of $568$ .

$x = \frac{(- 24 i \pm i \cdot \sqrt{4 (142)}) i}{4 i i}$

Step 2.3.2.2.12.2

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

$x = \frac{(- 24 i \pm i \cdot \sqrt{2^{2} \cdot 142}) i}{4 i i}$

Step 2.3.2.2.13

Pull terms out from under the radical.

$x = \frac{(- 24 i \pm i \cdot (2 \sqrt{142})) i}{4 i i}$

Step 2.3.2.2.14

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

$x = \frac{(- 24 i \pm 2 i \sqrt{142}) i}{4 i i}$

Step 2.3.2.3

Simplify the denominator.

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Step 2.3.2.3.1

Add parentheses.

$x = \frac{(- 24 i \pm 2 i \sqrt{142}) i}{4 (i i)}$

Step 2.3.2.3.2

Raise $i$ to the power of $1$ .

$x = \frac{(- 24 i \pm 2 i \sqrt{142}) i}{4 (i i)}$

Step 2.3.2.3.3

Raise $i$ to the power of $1$ .

$x = \frac{(- 24 i \pm 2 i \sqrt{142}) i}{4 (i i)}$

Step 2.3.2.3.4

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

$x = \frac{(- 24 i \pm 2 i \sqrt{142}) i}{4 i^{1 + 1}}$

Step 2.3.2.3.5

Add $1$ and $1$ .

$x = \frac{(- 24 i \pm 2 i \sqrt{142}) i}{4 i^{2}}$

Step 2.3.2.3.6

Rewrite $i^{2}$ as $- 1$ .

$x = \frac{(- 24 i \pm 2 i \sqrt{142}) i}{4 \cdot - 1}$

Step 2.3.3

Move the negative in front of the fraction.

$x = - \frac{(- 24 i \pm 2 i \sqrt{142}) i}{4}$

Step 2.3.4

Factor $- 1$ out of $- 24 i \pm 2 i \sqrt{142}$ .

$x = - \frac{- 1 (- (- 24 i \pm 2 i \sqrt{142})) i}{4}$

Step 2.3.5

Multiply $- 1$ by $- 1$ .

$x = - \frac{1 (- 24 i \pm 2 i \sqrt{142}) i}{4}$

Step 2.3.6

Multiply $- 24 i \pm 2 i \sqrt{142}$ by $1$ .

$x = - \frac{(- 24 i \pm 2 i \sqrt{142}) i}{4}$

Step 2.4

The final answer is the combination of both solutions.

$x = \frac{(24 i - 2 i \sqrt{142}) i}{4}, \frac{(24 i + 2 i \sqrt{142}) i}{4}$

$x = - \frac{(- 24 i \pm 2 i \sqrt{142}) i}{4}$

Step 3