Solve the following equation:

[tex]\[ x^2 + 4x + 5 = 0 \][/tex]

Use the quadratic formula:

[tex]\[ x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1} \][/tex]



Answer :

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.