Answer :
### Solution and Explanation:
Given the roots of the quadratic equation,
[tex]\[ x = \pm \frac{\sqrt{51}}{3} - 3i \][/tex]
Let's describe the method to solve and justify the quadratic equation.
### Step 1: Understanding the Nature of the Roots
The roots provided are in the form of complex numbers. This means that the quadratic equation has complex roots, which typically happens when the discriminant [tex]\(b^2 - 4ac\)[/tex] of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is negative.
### Step 2: Applying the Quadratic Formula
For any quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex], the roots can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
### Step 3: Identifying Components from Given Roots
The roots given are [tex]\(x = \pm \frac{\sqrt{51}}{3} - 3i\)[/tex]. Notice that they can be written in the form [tex]\(x = \alpha \pm \beta i\)[/tex] where:
- [tex]\(\alpha = -3\)[/tex]
- [tex]\(\beta = \frac{\sqrt{51}}{3}\)[/tex]
### Step 4: Forming the Quadratic Equation
If the roots of the quadratic equation are [tex]\(x = p \pm qi\)[/tex], then the equation can be formed by reversing the process:
[tex]\[ (x - (\alpha + \beta i))(x - (\alpha - \beta i)) = 0 \][/tex]
Given [tex]\(\alpha = -3\)[/tex] and [tex]\(\beta i = \pm \frac{\sqrt{51}}{3}\)[/tex]:
[tex]\[ (x - (-3 + \frac{\sqrt{51}}{3}i))(x - (-3 - \frac{\sqrt{51}}{3}i)) = 0 \][/tex]
Simplifying it:
[tex]\[ (x + 3 - \frac{\sqrt{51}}{3}i)(x + 3 + \frac{\sqrt{51}}{3}i) = 0 \][/tex]
### Step 5: Simplifying Further
This can be written as a product of sums and differences:
[tex]\[ \left[ (x + 3) - \frac{\sqrt{51}}{3}i \right] \left[ (x + 3) + \frac{\sqrt{51}}{3}i \right] = 0 \][/tex]
Using the formula for the difference of squares [tex]\((a + b)(a - b) = a^2 - b^2\)[/tex]:
Let [tex]\(u = x + 3\)[/tex] and [tex]\(v = \frac{\sqrt{51}}{3}i\)[/tex]:
[tex]\[ (u - v)(u + v) = u^2 - v^2 \][/tex]
Substituting back for [tex]\(u\)[/tex] and [tex]\(v\)[/tex]:
[tex]\[ \left( x + 3 \right)^2 - \left( \frac{\sqrt{51}}{3}i \right)^2 = 0 \][/tex]
### Step 6: Solving the Differences of Squares
Calculate [tex]\(v^2 = \left( \frac{\sqrt{51}}{3}i \right)^2\)[/tex]:
[tex]\[ \left( \frac{\sqrt{51}}{3}i \right)^2 = \left( \frac{\sqrt{51}}{3} \right)^2 \cdot i^2 = \frac{51}{9} \cdot (-1) = -\frac{51}{9} = -\frac{17}{3} \][/tex]
So,
[tex]\[ \left(x + 3\right)^2 - \left(-\frac{17}{3}\right) = 0 \][/tex]
Simplifying:
[tex]\[ \left(x + 3\right)^2 + \frac{17}{3} = 0 \][/tex]
### Conclusion
This quadratic equation confirms that the given complex roots
[tex]\[ x = \pm \frac{\sqrt{51}}{3} - 3i \][/tex]
are indeed the roots derived from the original quadratic equation.
### Justification
- Quadratic Formula: The quadratic formula was fundamental in understanding the general approach to finding roots.
- Roots Confirmation: The transformation from the roots back into the quadratic equation verifies their correctness.
- Complex Numbers: Given the nature of complex numbers, the discriminant [tex]\(b^2 - 4ac\)[/tex] being negative matches with provided roots.
The systematic approach and algebraic manipulations confirm the nature of the roots and properly justify the solution to the quadratic equation.
Given the roots of the quadratic equation,
[tex]\[ x = \pm \frac{\sqrt{51}}{3} - 3i \][/tex]
Let's describe the method to solve and justify the quadratic equation.
### Step 1: Understanding the Nature of the Roots
The roots provided are in the form of complex numbers. This means that the quadratic equation has complex roots, which typically happens when the discriminant [tex]\(b^2 - 4ac\)[/tex] of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is negative.
### Step 2: Applying the Quadratic Formula
For any quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex], the roots can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
### Step 3: Identifying Components from Given Roots
The roots given are [tex]\(x = \pm \frac{\sqrt{51}}{3} - 3i\)[/tex]. Notice that they can be written in the form [tex]\(x = \alpha \pm \beta i\)[/tex] where:
- [tex]\(\alpha = -3\)[/tex]
- [tex]\(\beta = \frac{\sqrt{51}}{3}\)[/tex]
### Step 4: Forming the Quadratic Equation
If the roots of the quadratic equation are [tex]\(x = p \pm qi\)[/tex], then the equation can be formed by reversing the process:
[tex]\[ (x - (\alpha + \beta i))(x - (\alpha - \beta i)) = 0 \][/tex]
Given [tex]\(\alpha = -3\)[/tex] and [tex]\(\beta i = \pm \frac{\sqrt{51}}{3}\)[/tex]:
[tex]\[ (x - (-3 + \frac{\sqrt{51}}{3}i))(x - (-3 - \frac{\sqrt{51}}{3}i)) = 0 \][/tex]
Simplifying it:
[tex]\[ (x + 3 - \frac{\sqrt{51}}{3}i)(x + 3 + \frac{\sqrt{51}}{3}i) = 0 \][/tex]
### Step 5: Simplifying Further
This can be written as a product of sums and differences:
[tex]\[ \left[ (x + 3) - \frac{\sqrt{51}}{3}i \right] \left[ (x + 3) + \frac{\sqrt{51}}{3}i \right] = 0 \][/tex]
Using the formula for the difference of squares [tex]\((a + b)(a - b) = a^2 - b^2\)[/tex]:
Let [tex]\(u = x + 3\)[/tex] and [tex]\(v = \frac{\sqrt{51}}{3}i\)[/tex]:
[tex]\[ (u - v)(u + v) = u^2 - v^2 \][/tex]
Substituting back for [tex]\(u\)[/tex] and [tex]\(v\)[/tex]:
[tex]\[ \left( x + 3 \right)^2 - \left( \frac{\sqrt{51}}{3}i \right)^2 = 0 \][/tex]
### Step 6: Solving the Differences of Squares
Calculate [tex]\(v^2 = \left( \frac{\sqrt{51}}{3}i \right)^2\)[/tex]:
[tex]\[ \left( \frac{\sqrt{51}}{3}i \right)^2 = \left( \frac{\sqrt{51}}{3} \right)^2 \cdot i^2 = \frac{51}{9} \cdot (-1) = -\frac{51}{9} = -\frac{17}{3} \][/tex]
So,
[tex]\[ \left(x + 3\right)^2 - \left(-\frac{17}{3}\right) = 0 \][/tex]
Simplifying:
[tex]\[ \left(x + 3\right)^2 + \frac{17}{3} = 0 \][/tex]
### Conclusion
This quadratic equation confirms that the given complex roots
[tex]\[ x = \pm \frac{\sqrt{51}}{3} - 3i \][/tex]
are indeed the roots derived from the original quadratic equation.
### Justification
- Quadratic Formula: The quadratic formula was fundamental in understanding the general approach to finding roots.
- Roots Confirmation: The transformation from the roots back into the quadratic equation verifies their correctness.
- Complex Numbers: Given the nature of complex numbers, the discriminant [tex]\(b^2 - 4ac\)[/tex] being negative matches with provided roots.
The systematic approach and algebraic manipulations confirm the nature of the roots and properly justify the solution to the quadratic equation.
