To solve the quadratic equation [tex]\( x^2 + 4x + 5 = 0 \)[/tex], we can use the quadratic formula, which is:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
Here, the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( c = 5 \)[/tex]
Let's follow the steps to find the roots of this equation using the quadratic formula.
1. Calculate the Discriminant
The discriminant [tex]\( \Delta \)[/tex] of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = 5 \)[/tex]:
[tex]\[
\Delta = 4^2 - 4 \cdot 1 \cdot 5
\][/tex]
[tex]\[
\Delta = 16 - 20
\][/tex]
[tex]\[
\Delta = -4
\][/tex]
The discriminant is [tex]\(-4\)[/tex], which is negative. This indicates that the quadratic equation has two complex (non-real) roots.
2. Apply the Quadratic Formula
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and the discriminant [tex]\( \Delta \)[/tex] into the quadratic formula:
[tex]\[
x = \frac{{-b \pm \sqrt{\Delta}}}{2a}
\][/tex]
Since [tex]\( \Delta = -4 \)[/tex]:
[tex]\[
x = \frac{{-4 \pm \sqrt{{-4}}}}{2 \cdot 1}
\][/tex]
3. Simplify the Expression
The square root of [tex]\(-4\)[/tex] can be written as [tex]\(2i\)[/tex] (where [tex]\( i \)[/tex] is the imaginary unit, with [tex]\( i^2 = -1 \)[/tex]):
[tex]\[
\sqrt{-4} = \sqrt{(-1) \cdot 4} = \sqrt{-1} \cdot \sqrt{4} = 2i
\][/tex]
Now substitute it back into the formula:
[tex]\[
x = \frac{{-4 \pm 2i}}{2}
\][/tex]
4. Separate the Two Roots
[tex]\[
x = \frac{{-4 + 2i}}{2} \quad \text{and} \quad x = \frac{{-4 - 2i}}{2}
\][/tex]
Simplify each term:
[tex]\[
x = -2 + i \quad \text{and} \quad x = -2 - i
\][/tex]
So, the roots of the quadratic equation [tex]\( x^2 + 4x + 5 = 0 \)[/tex] are:
[tex]\[
x_1 = -2 + i
\][/tex]
[tex]\[
x_2 = -2 - i
\][/tex]
These are the two complex conjugate roots of the given quadratic equation.
