Precalculus Examples

$3 - 7 i$

Step 1

Roots are the points where the graph intercepts with the x-axis $(y = 0)$ .

$y = 0$ at the roots

Step 2

The root at $x = 3 - 7 i$ was found by solving for $x$ when $x - (3 - 7 i) = y$ and $y = 0$ .

The factor is $x - 3 + 7 i$

Step 3

The root at $x = 3 + 7 i$ was found by solving for $x$ when $x - (3 + 7 i) = y$ and $y = 0$ .

The factor is $x - 3 - 7 i$

Step 4

Combine all the factors into a single equation.

$y = (x - 3 + 7 i) (x - 3 - 7 i)$

Step 5

Multiply all the factors to simplify the equation $y = (x - 3 + 7 i) (x - 3 - 7 i)$ .

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Step 5.1

Expand $(x - 3 + 7 i) (x - 3 - 7 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x \cdot x + x \cdot - 3 + x (- 7 i) - 3 x - 3 \cdot - 3 - 3 (- 7 i) + 7 i x + 7 i \cdot - 3 + 7 i (- 7 i)$

Step 5.2

Simplify terms.

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Step 5.2.1

Combine the opposite terms in $x \cdot x + x \cdot - 3 + x (- 7 i) - 3 x - 3 \cdot - 3 - 3 (- 7 i) + 7 i x + 7 i \cdot - 3 + 7 i (- 7 i)$ .

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Step 5.2.1.1

Reorder the factors in the terms $x (- 7 i)$ and $7 i x$ .

$y = x \cdot x + x \cdot - 3 - 7 i x - 3 x - 3 \cdot - 3 - 3 (- 7 i) + 7 i x + 7 i \cdot - 3 + 7 i (- 7 i)$

Step 5.2.1.2

Add $- 7 i x$ and $7 i x$ .

$y = x \cdot x + x \cdot - 3 + 0 - 3 x - 3 \cdot - 3 - 3 (- 7 i) + 7 i \cdot - 3 + 7 i (- 7 i)$

Step 5.2.1.3

Add $x \cdot x + x \cdot - 3$ and $0$ .

$y = x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 - 3 (- 7 i) + 7 i \cdot - 3 + 7 i (- 7 i)$

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

Multiply $x$ by $x$ .

$y = x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3 - 3 (- 7 i) + 7 i \cdot - 3 + 7 i (- 7 i)$

Step 5.2.2.2

Move $- 3$ to the left of $x$ .

$y = x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3 - 3 (- 7 i) + 7 i \cdot - 3 + 7 i (- 7 i)$

Step 5.2.2.3

Multiply $- 3$ by $- 3$ .

$y = x^{2} - 3 x - 3 x + 9 - 3 (- 7 i) + 7 i \cdot - 3 + 7 i (- 7 i)$

Step 5.2.2.4

Multiply $- 7$ by $- 3$ .

$y = x^{2} - 3 x - 3 x + 9 + 21 i + 7 i \cdot - 3 + 7 i (- 7 i)$

Step 5.2.2.5

Multiply $- 3$ by $7$ .

$y = x^{2} - 3 x - 3 x + 9 + 21 i - 21 i + 7 i (- 7 i)$

Step 5.2.2.6

Multiply $7 i (- 7 i)$ .

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Step 5.2.2.6.1

Multiply $- 7$ by $7$ .

$y = x^{2} - 3 x - 3 x + 9 + 21 i - 21 i - 49 i i$

Step 5.2.2.6.2

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

$y = x^{2} - 3 x - 3 x + 9 + 21 i - 21 i - 49 (i i)$

Step 5.2.2.6.3

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

$y = x^{2} - 3 x - 3 x + 9 + 21 i - 21 i - 49 (i i)$

Step 5.2.2.6.4

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

$y = x^{2} - 3 x - 3 x + 9 + 21 i - 21 i - 49 i^{1 + 1}$

Step 5.2.2.6.5

Add $1$ and $1$ .

$y = x^{2} - 3 x - 3 x + 9 + 21 i - 21 i - 49 i^{2}$

Step 5.2.2.7

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

$y = x^{2} - 3 x - 3 x + 9 + 21 i - 21 i - 49 \cdot - 1$

Step 5.2.2.8

Multiply $- 49$ by $- 1$ .

$y = x^{2} - 3 x - 3 x + 9 + 21 i - 21 i + 49$

Step 5.2.3

Simplify by adding terms.

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Step 5.2.3.1

Combine the opposite terms in $x^{2} - 3 x - 3 x + 9 + 21 i - 21 i + 49$ .

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Step 5.2.3.1.1

Subtract $21 i$ from $21 i$ .

$y = x^{2} - 3 x - 3 x + 9 + 0 + 49$

Step 5.2.3.1.2

Add $x^{2} - 3 x - 3 x + 9$ and $0$ .

$y = x^{2} - 3 x - 3 x + 9 + 49$

Step 5.2.3.2

Subtract $3 x$ from $- 3 x$ .

$y = x^{2} - 6 x + 9 + 49$

Step 5.2.3.3

Add $9$ and $49$ .

$y = x^{2} - 6 x + 58$

Step 6