To solve the quadratic equation [tex]\(x^2 + 5x + 9 = 0\)[/tex] using the quadratic formula, we'll follow these detailed steps:
### Quadratic Formula
The quadratic formula is given by:
[tex]\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\][/tex]
where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the coefficients of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex].
### Identifying Coefficients
For the given equation [tex]\(x^2 + 5x + 9 = 0\)[/tex], we identify the coefficients as follows:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = 5\)[/tex]
- [tex]\(c = 9\)[/tex]
### Calculating the Discriminant
The discriminant of a quadratic equation is the part under the square root in the quadratic formula, given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = 5^2 - 4 \cdot 1 \cdot 9 \][/tex]
[tex]\[ \Delta = 25 - 36 \][/tex]
[tex]\[ \Delta = -11 \][/tex]
### Discriminant Analysis
Since the discriminant [tex]\(\Delta = -11\)[/tex] is less than zero, this indicates that the quadratic equation has no real roots; instead, it has two complex conjugate roots.
### Calculating the Roots
To find the complex roots, we use the quadratic formula while acknowledging that the square root of a negative number involves imaginary numbers ([tex]\(i\)[/tex]), where [tex]\(i = \sqrt{-1}\)[/tex].
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex]:
[tex]\[ x = \frac{-5 \pm \sqrt{-11}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{-5 \pm \sqrt{11}i}{2} \][/tex]
### Simplifying the Complex Roots
The roots can be written as:
[tex]\[ x = \frac{-5 + \sqrt{11}i}{2} \][/tex]
[tex]\[ x = \frac{-5 - \sqrt{11}i}{2} \][/tex]
These are the two complex roots of the given equation:
### Conclusion
Given the context of the question and the options provided:
a. 4 and 1
b. -5 and -4
c. -4 and -1
d. 5 and 4
None of the options provided are correct, as the actual roots are complex numbers. So, the correct understanding is that [tex]\(x^2 + 5x + 9 = 0\)[/tex] has complex roots, not real roots.
