To find the quadratic equation given the roots [tex]\( x = 2 + i \)[/tex] and [tex]\( x = 2 - i \)[/tex], we can use the fact that for a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], the sum of the roots [tex]\( (x_1 + x_2) \)[/tex] is equal to [tex]\( -b/a \)[/tex] and the product of the roots [tex]\( (x_1 \cdot x_2) \)[/tex] is equal to [tex]\( c/a \)[/tex].
Given:
- Roots [tex]\( x_1 = 2 + i \)[/tex] and [tex]\( x_2 = 2 - i \)[/tex]
Step-by-step solution:
1. Sum of the roots:
Sum [tex]\( (x_1 + x_2) = (2 + i) + (2 - i) = 4 \)[/tex]
Therefore, [tex]\( -b/a = 4 \)[/tex].
2. Product of the roots:
Product [tex]\( (x_1 \cdot x_2)= (2 + i)(2 - i) = 2^2 - i^2 = 4 - (-1) = 4 + 1 = 5 \)[/tex]
Therefore, [tex]\( c/a = 5 \)[/tex].
3. Considering the standard form of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], and typically [tex]\( a = 1 \)[/tex] for simplicity (which does not change the equation’s nature), we can write:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -4 \)[/tex] (from [tex]\( -b = 4 \)[/tex] hence [tex]\( b = -4 \)[/tex])
- [tex]\( c = 5 \)[/tex] (from [tex]\( c = 5 \)[/tex])
Putting these values into the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex]:
[tex]\[
1x^2 - 4x + 5 = 0
\][/tex]
Therefore, the quadratic equation with the roots [tex]\( x = 2 \pm i \)[/tex] is:
[tex]\[ x^2 - 4x + 5 = 0 \][/tex]
