To find the coordinates of the roots of the equation [tex]\(x^2 + 4x + 3 = 0\)[/tex], we need to use the quadratic formula, which is given by:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
Here, the coefficients of the quadratic equation are:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = 4\)[/tex]
- [tex]\(c = 3\)[/tex]
First, we calculate the discriminant ([tex]\(\Delta\)[/tex]) using the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], we get:
[tex]\[
\Delta = 4^2 - 4 \cdot 1 \cdot 3 = 16 - 12 = 4
\][/tex]
Next, we calculate the roots using the quadratic formula. There are two roots, corresponding to the [tex]\(+\)[/tex] and [tex]\(-\)[/tex] in the quadratic formula.
For the first root ([tex]\(x_1\)[/tex]):
[tex]\[
x_1 = \frac{{-b + \sqrt{\Delta}}}{2a}
\][/tex]
Substituting the values, we get:
[tex]\[
x_1 = \frac{{-4 + \sqrt{4}}}{2 \cdot 1} = \frac{{-4 + 2}}{2} = \frac{{-2}}{2} = -1
\][/tex]
For the second root ([tex]\(x_2\)[/tex]):
[tex]\[
x_2 = \frac{{-b - \sqrt{\Delta}}}{2a}
\][/tex]
Substituting the values, we get:
[tex]\[
x_2 = \frac{{-4 - \sqrt{4}}}{2 \cdot 1} = \frac{{-4 - 2}}{2} = \frac{{-6}}{2} = -3
\][/tex]
Therefore, the coordinates of the roots of the equation [tex]\(x^2 + 4x + 3 = 0\)[/tex] are:
[tex]\[
x_1 = -1 \quad \text{and} \quad x_2 = -3
\][/tex]
