Algebra Examples

- 5 + 2 i

$- 5 + 2 i$ ,

- 5 i

$- 5 i$

Step 1

x = - 5 + 2 i

$x = - 5 + 2 i$ and

x = - 5 i

$x = - 5 i$ are the two real distinct solutions for the quadratic equation, which means that

x - - 5 + 2 i

$x - - 5 + 2 i$ and

x - (- 5 i)

$x - (- 5 i)$ are the factors of the quadratic equation.

(x + 5 + 2 i) (x + 5 i) = 0

$(x + 5 + 2 i) (x + 5 i) = 0$

Step 2

Expand

(x + 5 + 2 i) (x + 5 i)

$(x + 5 + 2 i) (x + 5 i)$ by multiplying each term in the first expression by each term in the second expression.

x \cdot x + x (5 i) + 5 x + 5 (5 i) + 2 i x + 2 i (5 i) = 0

$x \cdot x + x (5 i) + 5 x + 5 (5 i) + 2 i x + 2 i (5 i) = 0$

Step 3

Simplify each term.

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Step 3.1

Multiply

x

$x$ by

x

$x$ .

x^{2} + x (5 i) + 5 x + 5 (5 i) + 2 i x + 2 i (5 i) = 0

$x^{2} + x (5 i) + 5 x + 5 (5 i) + 2 i x + 2 i (5 i) = 0$

Step 3.2

Move

5

$5$ to the left of

x

$x$ .

x^{2} + 5 \cdot (x i) + 5 x + 5 (5 i) + 2 i x + 2 i (5 i) = 0

$x^{2} + 5 \cdot (x i) + 5 x + 5 (5 i) + 2 i x + 2 i (5 i) = 0$

Step 3.3

Multiply

5

$5$ by

5

$5$ .

x^{2} + 5 x i + 5 x + 25 i + 2 i x + 2 i (5 i) = 0

$x^{2} + 5 x i + 5 x + 25 i + 2 i x + 2 i (5 i) = 0$

Step 3.4

Multiply

2 i (5 i)

$2 i (5 i)$ .

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Step 3.4.1

Multiply

5

$5$ by

2

$2$ .

x^{2} + 5 x i + 5 x + 25 i + 2 i x + 10 i i = 0

$x^{2} + 5 x i + 5 x + 25 i + 2 i x + 10 i i = 0$

Step 3.4.2

Raise

i

$i$ to the power of

1

$1$ .

x^{2} + 5 x i + 5 x + 25 i + 2 i x + 10 (i i) = 0

$x^{2} + 5 x i + 5 x + 25 i + 2 i x + 10 (i i) = 0$

Step 3.4.3

Raise

i

$i$ to the power of

1

$1$ .

x^{2} + 5 x i + 5 x + 25 i + 2 i x + 10 (i i) = 0

$x^{2} + 5 x i + 5 x + 25 i + 2 i x + 10 (i i) = 0$

Step 3.4.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

x^{2} + 5 x i + 5 x + 25 i + 2 i x + 10 i^{1 + 1} = 0

$x^{2} + 5 x i + 5 x + 25 i + 2 i x + 10 i^{1 + 1} = 0$

Step 3.4.5

Add

1

$1$ and

1

$1$ .

x^{2} + 5 x i + 5 x + 25 i + 2 i x + 10 i^{2} = 0

$x^{2} + 5 x i + 5 x + 25 i + 2 i x + 10 i^{2} = 0$

x^{2} + 5 x i + 5 x + 25 i + 2 i x + 10 i^{2} = 0

$x^{2} + 5 x i + 5 x + 25 i + 2 i x + 10 i^{2} = 0$

Step 3.5

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

x^{2} + 5 x i + 5 x + 25 i + 2 i x + 10 \cdot - 1 = 0

$x^{2} + 5 x i + 5 x + 25 i + 2 i x + 10 \cdot - 1 = 0$

Step 3.6

Multiply

10

$10$ by

- 1

$- 1$ .

x^{2} + 5 x i + 5 x + 25 i + 2 i x - 10 = 0

$x^{2} + 5 x i + 5 x + 25 i + 2 i x - 10 = 0$

x^{2} + 5 x i + 5 x + 25 i + 2 i x - 10 = 0

$x^{2} + 5 x i + 5 x + 25 i + 2 i x - 10 = 0$

Step 4

Add

5 x i

$5 x i$ and

2 i x

$2 i x$ .

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Step 4.1

Move

i

$i$ .

x^{2} + 5 x i + 2 x i + 5 x + 25 i - 10 = 0

$x^{2} + 5 x i + 2 x i + 5 x + 25 i - 10 = 0$

Step 4.2

Add

5 x i

$5 x i$ and

2 x i

$2 x i$ .

x^{2} + 7 x i + 5 x + 25 i - 10 = 0

$x^{2} + 7 x i + 5 x + 25 i - 10 = 0$

x^{2} + 7 x i + 5 x + 25 i - 10 = 0

$x^{2} + 7 x i + 5 x + 25 i - 10 = 0$

Step 5

The standard quadratic equation using the given set of solutions

{- 5 + 2 i, - 5 i}

${- 5 + 2 i, - 5 i}$ is

y = x^{2} + 7 x i + 5 x + 25 i - 10

$y = x^{2} + 7 x i + 5 x + 25 i - 10$ .

y = x^{2} + 7 x i + 5 x + 25 i - 10

$y = x^{2} + 7 x i + 5 x + 25 i - 10$

Step 6