Find the Quadratic Equation 1+5i , 1-5i

$1 + 5 i$ , $1 - 5 i$

Step 1

$x = 1 + 5 i$ and $x = 1 - 5 i$ are the two real distinct solutions for the quadratic equation, which means that $x - 1 + 5 i$ and $x - 1 - 5 i$ are the factors of the quadratic equation.

$(x - 1 + 5 i) (x - 1 - 5 i) = 0$

Step 2

Expand $(x - 1 + 5 i) (x - 1 - 5 i)$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x + x \cdot - 1 + x (- 5 i) - 1 x - 1 \cdot - 1 - 1 (- 5 i) + 5 i x + 5 i \cdot - 1 + 5 i (- 5 i) = 0$

Step 3

Simplify terms.

Combine the opposite terms in $x \cdot x + x \cdot - 1 + x (- 5 i) - 1 x - 1 \cdot - 1 - 1 (- 5 i) + 5 i x + 5 i \cdot - 1 + 5 i (- 5 i)$ .

Reorder the factors in the terms $x (- 5 i)$ and $5 i x$ .

$x \cdot x + x \cdot - 1 - 5 i x - 1 x - 1 \cdot - 1 - 1 (- 5 i) + 5 i x + 5 i \cdot - 1 + 5 i (- 5 i) = 0$

Add $- 5 i x$ and $5 i x$ .

$x \cdot x + x \cdot - 1 + 0 - 1 x - 1 \cdot - 1 - 1 (- 5 i) + 5 i \cdot - 1 + 5 i (- 5 i) = 0$

Add $x \cdot x + x \cdot - 1$ and $0$ .

$x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 - 1 (- 5 i) + 5 i \cdot - 1 + 5 i (- 5 i) = 0$

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1 - 1 (- 5 i) + 5 i \cdot - 1 + 5 i (- 5 i) = 0$

Move $- 1$ to the left of $x$ .

$x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1 - 1 (- 5 i) + 5 i \cdot - 1 + 5 i (- 5 i) = 0$

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x - 1 x - 1 \cdot - 1 - 1 (- 5 i) + 5 i \cdot - 1 + 5 i (- 5 i) = 0$

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x - x - 1 \cdot - 1 - 1 (- 5 i) + 5 i \cdot - 1 + 5 i (- 5 i) = 0$

Multiply $- 1$ by $- 1$ .

$x^{2} - x - x + 1 - 1 (- 5 i) + 5 i \cdot - 1 + 5 i (- 5 i) = 0$

Multiply $- 5$ by $- 1$ .

$x^{2} - x - x + 1 + 5 i + 5 i \cdot - 1 + 5 i (- 5 i) = 0$

Multiply $- 1$ by $5$ .

$x^{2} - x - x + 1 + 5 i - 5 i + 5 i (- 5 i) = 0$

Multiply $5 i (- 5 i)$ .

Multiply $- 5$ by $5$ .

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

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

$x^{2} - x - x + 1 + 5 i - 5 i - 25 (i i) = 0$

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

$x^{2} - x - x + 1 + 5 i - 5 i - 25 (i i) = 0$

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 25$ by $- 1$ .

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

Simplify by adding terms.

Combine the opposite terms in $x^{2} - x - x + 1 + 5 i - 5 i + 25$ .

Subtract $5 i$ from $5 i$ .

$x^{2} - x - x + 1 + 0 + 25 = 0$

Add $x^{2} - x - x + 1$ and $0$ .

$x^{2} - x - x + 1 + 25 = 0$

Subtract $x$ from $- x$ .

$x^{2} - 2 x + 1 + 25 = 0$

Add $1$ and $25$ .

$x^{2} - 2 x + 26 = 0$

Step 4

The standard quadratic equation using the given set of solutions ${1 + 5 i, 1 - 5 i}$ is $y = x^{2} - 2 x + 26$ .

$y = x^{2} - 2 x + 26$

Step 5

Find the Quadratic Equation 1+5i , 1-5i

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