Algebra Examples

$x^{2} - 7 x - 5 i x - 15 i + 12$

Step 1

Set $x^{2} - 7 x - 5 i x - 15 i + 12$ equal to $0$ .

$x^{2} - 7 x - 5 i x - 15 i + 12 = 0$

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 $a = 1$ , $b = - 7 - 5 i$ , and $c = 12 - 15 i$ into the quadratic formula and solve for $x$ .

$\frac{- (- 7 - 5 i) \pm \sqrt{{(- 7 - 5 i)}^{2} - 4 \cdot (1 \cdot (12 - 15 i))}}{2 \cdot 1}$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the numerator.

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Step 2.3.1.1

Apply the distributive property.

$x = \frac{7 - (- 5 i) \pm \sqrt{{(- 7 - 5 i)}^{2} - 4 \cdot 1 \cdot (12 - 15 i)}}{2 \cdot 1}$

Step 2.3.1.2

Multiply $- 1$ by $- 7$ .

$x = \frac{7 - (- 5 i) \pm \sqrt{{(- 7 - 5 i)}^{2} - 4 \cdot 1 \cdot (12 - 15 i)}}{2 \cdot 1}$

Step 2.3.1.3

Multiply $- 5$ by $- 1$ .

$x = \frac{7 + 5 i \pm \sqrt{{(- 7 - 5 i)}^{2} - 4 \cdot 1 \cdot (12 - 15 i)}}{2 \cdot 1}$

Step 2.3.1.4

Rewrite ${(- 7 - 5 i)}^{2}$ as $(- 7 - 5 i) (- 7 - 5 i)$ .

$x = \frac{7 + 5 i \pm \sqrt{(- 7 - 5 i) (- 7 - 5 i) - 4 \cdot 1 \cdot (12 - 15 i)}}{2 \cdot 1}$

Step 2.3.1.5

Expand $(- 7 - 5 i) (- 7 - 5 i)$ using the FOIL Method.

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Step 2.3.1.5.1

Apply the distributive property.

$x = \frac{7 + 5 i \pm \sqrt{- 7 (- 7 - 5 i) - 5 i (- 7 - 5 i) - 4 \cdot 1 \cdot (12 - 15 i)}}{2 \cdot 1}$

Step 2.3.1.5.2

Apply the distributive property.

$x = \frac{7 + 5 i \pm \sqrt{- 7 \cdot - 7 - 7 (- 5 i) - 5 i (- 7 - 5 i) - 4 \cdot 1 \cdot (12 - 15 i)}}{2 \cdot 1}$

Step 2.3.1.5.3

Apply the distributive property.

$x = \frac{7 + 5 i \pm \sqrt{- 7 \cdot - 7 - 7 (- 5 i) - 5 i \cdot - 7 - 5 i (- 5 i) - 4 \cdot 1 \cdot (12 - 15 i)}}{2 \cdot 1}$

Step 2.3.1.6

Simplify and combine like terms.

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Step 2.3.1.6.1

Simplify each term.

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Step 2.3.1.6.1.1

Multiply $- 7$ by $- 7$ .

$x = \frac{7 + 5 i \pm \sqrt{49 - 7 (- 5 i) - 5 i \cdot - 7 - 5 i (- 5 i) - 4 \cdot 1 \cdot (12 - 15 i)}}{2 \cdot 1}$

Step 2.3.1.6.1.2

Multiply $- 5$ by $- 7$ .

$x = \frac{7 + 5 i \pm \sqrt{49 + 35 i - 5 i \cdot - 7 - 5 i (- 5 i) - 4 \cdot 1 \cdot (12 - 15 i)}}{2 \cdot 1}$

Step 2.3.1.6.1.3

Multiply $- 7$ by $- 5$ .

$x = \frac{7 + 5 i \pm \sqrt{49 + 35 i + 35 i - 5 i (- 5 i) - 4 \cdot 1 \cdot (12 - 15 i)}}{2 \cdot 1}$

Step 2.3.1.6.1.4

Multiply $- 5 i (- 5 i)$ .

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Step 2.3.1.6.1.4.1

Multiply $- 5$ by $- 5$ .

$x = \frac{7 + 5 i \pm \sqrt{49 + 35 i + 35 i + 25 i i - 4 \cdot 1 \cdot (12 - 15 i)}}{2 \cdot 1}$

Step 2.3.1.6.1.4.2

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

$x = \frac{7 + 5 i \pm \sqrt{49 + 35 i + 35 i + 25 (i i) - 4 \cdot 1 \cdot (12 - 15 i)}}{2 \cdot 1}$

Step 2.3.1.6.1.4.3

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

$x = \frac{7 + 5 i \pm \sqrt{49 + 35 i + 35 i + 25 (i i) - 4 \cdot 1 \cdot (12 - 15 i)}}{2 \cdot 1}$

Step 2.3.1.6.1.4.4

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

$x = \frac{7 + 5 i \pm \sqrt{49 + 35 i + 35 i + 25 i^{1 + 1} - 4 \cdot 1 \cdot (12 - 15 i)}}{2 \cdot 1}$

Step 2.3.1.6.1.4.5

Add $1$ and $1$ .

$x = \frac{7 + 5 i \pm \sqrt{49 + 35 i + 35 i + 25 i^{2} - 4 \cdot 1 \cdot (12 - 15 i)}}{2 \cdot 1}$

Step 2.3.1.6.1.5

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

$x = \frac{7 + 5 i \pm \sqrt{49 + 35 i + 35 i + 25 \cdot - 1 - 4 \cdot 1 \cdot (12 - 15 i)}}{2 \cdot 1}$

Step 2.3.1.6.1.6

Multiply $25$ by $- 1$ .

$x = \frac{7 + 5 i \pm \sqrt{49 + 35 i + 35 i - 25 - 4 \cdot 1 \cdot (12 - 15 i)}}{2 \cdot 1}$

Step 2.3.1.6.2

Subtract $25$ from $49$ .

$x = \frac{7 + 5 i \pm \sqrt{24 + 35 i + 35 i - 4 \cdot 1 \cdot (12 - 15 i)}}{2 \cdot 1}$

Step 2.3.1.6.3

Add $35 i$ and $35 i$ .

$x = \frac{7 + 5 i \pm \sqrt{24 + 70 i - 4 \cdot 1 \cdot (12 - 15 i)}}{2 \cdot 1}$

Step 2.3.1.7

Multiply $- 4$ by $1$ .

$x = \frac{7 + 5 i \pm \sqrt{24 + 70 i - 4 \cdot (12 - 15 i)}}{2 \cdot 1}$

Step 2.3.1.8

Apply the distributive property.

$x = \frac{7 + 5 i \pm \sqrt{24 + 70 i - 4 \cdot 12 - 4 (- 15 i)}}{2 \cdot 1}$

Step 2.3.1.9

Multiply $- 4$ by $12$ .

$x = \frac{7 + 5 i \pm \sqrt{24 + 70 i - 48 - 4 (- 15 i)}}{2 \cdot 1}$

Step 2.3.1.10

Multiply $- 15$ by $- 4$ .

$x = \frac{7 + 5 i \pm \sqrt{24 + 70 i - 48 + 60 i}}{2 \cdot 1}$

Step 2.3.1.11

Subtract $48$ from $24$ .

$x = \frac{7 + 5 i \pm \sqrt{- 24 + 70 i + 60 i}}{2 \cdot 1}$

Step 2.3.1.12

Add $70 i$ and $60 i$ .

$x = \frac{7 + 5 i \pm \sqrt{- 24 + 130 i}}{2 \cdot 1}$

Step 2.3.2

Multiply $2$ by $1$ .

$x = \frac{7 + 5 i \pm \sqrt{- 24 + 130 i}}{2}$

Step 2.4

The final answer is the combination of both solutions.

$x = \frac{7 + 5 i + \sqrt{- 24 + 130 i}}{2}, \frac{7 + 5 i - \sqrt{- 24 + 130 i}}{2}$

$x = \frac{7 + 5 i \pm \sqrt{- 24 + 130 i}}{2}$

Step 3