Solve the following quadratic equation: `\ x^2-(2+i)x-(1-7i)=0`

Explanation

To solve the quadratic equation \( x^2 - (2+i)x - (1-7i) = 0 \), we will follow these steps:

### Step 1: Identify coefficients
The given quadratic equation is in the form \( ax^2 + bx + c = 0 \).
Here, we can identify:
- \( a = 1 \)
- \( b = -(2+i) = -2 - i \)
- \( c = -(1-7i) = -1 + 7i \)

### Step 2: Calculate the discriminant
The discriminant \( D \) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-2-i)^2 - 4(1)(-1 + 7i) \]

Calculating \( (-2-i)^2 \):
\[ (-2-i)^2 = (-2)^2 + 2(-2)(-i) + (-i)^2 = 4 + 4i - 1 = 3 + 4i \]

Now calculating \( -4(1)(-1 + 7i) \):
\[ -4(-1 + 7i) = 4 - 28i \]

Now substituting back into the discriminant:
\[ D = (3 + 4i) + (4 - 28i) = 7 - 24i \]

### Step 3: Calculate the roots using the quadratic formula
The roots of the quadratic equation are given by:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting \( a = 1 \) and \( b = -2 - i \):
\[ x = \frac{2 + i \pm \sqrt{7 - 24i}}{2} \]

### Step 4: Simplify \( \sqrt{7 - 24i} \)
Assume \( \sqrt{7 - 24i} = a + bi \). Then squaring both sides gives:
\[ 7 - 24i = (a + bi)^2 = a^2 + 2abi - b^2 \]
This leads to two equations:
1. \( a^2 - b^2 = 7 \)
2. \( 2ab = -24 \)

From the second equation, we can express \( ab \):
\[ ab = -12 \quad \Rightarrow \quad b = \frac{-12}{a} \]

Substituting \( b \) into the first equation:
\[ a^2 - \left(\frac{-12}{a}\right)^2 = 7 \]
This simplifies to:
\[ a^2 - \frac{144}{a^2} = 7 \]
Multiplying through by \( a^2 \):
\[ a^4 - 7a^2 - 144 = 0 \]
Letting \( y = a^2 \), we have:
\[ y^2 - 7y - 144 = 0 \]

### Step 5: Solve the quadratic equation for \( y \)
Using the quadratic formula:
\[ y = \frac{7 \pm \sqrt{(-7)^2 - 4(1)(-144)}}{2(1)} = \frac{7 \pm \sqrt{49 + 576}}{2} = \frac{7 \pm \sqrt{625}}{2} = \frac{7 \pm 25}{2} \]
Thus:
\[ y_1 = \frac{32}{2} = 16 \quad \text{and} \quad y_2 = \frac{-18}{2} = -9 \quad (\text{not valid since } y = a^2 \geq 0) \]
So, \( a^2 = 16 \) implies \( a = 4 \) or \( a = -4 \).

### Step 6: Find \( b \)
Using \( ab = -12 \):
If \( a = 4 \), then:
\[ 4b = -12 \quad \Rightarrow \quad b = -3 \]
Thus, \( \sqrt{7 - 24i} = 4 - 3i \).

### Step 7: Substitute back to find \( x \)
Now substituting back into the roots:
\[ x = \frac{2 + i \pm (4 - 3i)}{2} \]
Calculating the two roots:
1. \( x_1 = \frac{(2 + i) + (4 - 3i)}{2} = \frac{6 - 2i}{2} = 3 - i \)
2. \( x_2 = \frac{(2 + i) - (4 - 3i)}{2} = \frac{-2 + 4i}{2} = -1 + 2i \)

### Final Answer
The roots of the quadratic equation are:
\[ x_1 = 3 - i \quad \text{and} \quad x_2 = -1 + 2i \]