We're dealing with a quadratic equation of the form ax^2 + bx + c = 0. The coefficients of the equation, in this case, are a = 1, b = 13, and c = 9.
The roots of the quadratic equation can be found using the formula: x = [-b ± sqrt(b^2 - 4ac)] / 2a.
The term (b^2 - 4ac) underneath the square root is called the discriminant. It gives us information about the roots of the equation.
First, we calculate the discriminant:
D = b^2 - 4*a*c
= (13)^2 - 4*1*9
= 169 - 36
= 133
The discriminant is positive, so the equation has two distinct real roots.
Now let's find the roots:
For the first root, we have: x1 = [-b + sqrt(D)] / 2a
= [-13 + sqrt(133)] / (2*1)
= -0.734 (rounded to three decimal places)
For the second root, we have: x2 = [-b - sqrt(D)] / 2a
= [-13 - sqrt(133)] / (2*1)
= -12.266 (rounded to three decimal places)
So the roots of the given quadratic equation are -0.734 and -12.266.