To solve the quadratic equation [tex]\(x^2 - x - 90 = 0\)[/tex], we need to use the quadratic formula, which is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
In this specific equation, the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -1 \)[/tex]
- [tex]\( c = -90 \)[/tex]
First, we calculate the discriminant, [tex]\(\Delta\)[/tex], as follows:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = (-1)^2 - 4 \cdot 1 \cdot (-90) \][/tex]
[tex]\[ \Delta = 1 + 360 \][/tex]
[tex]\[ \Delta = 361 \][/tex]
Since the discriminant is positive, we will have two distinct real roots. We can find the solutions [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex] by plugging the values into the quadratic formula:
[tex]\[ x_1 = \frac{-(-1) + \sqrt{361}}{2 \cdot 1} \][/tex]
[tex]\[ x_1 = \frac{1 + 19}{2} \][/tex]
[tex]\[ x_1 = \frac{20}{2} \][/tex]
[tex]\[ x_1 = 10 \][/tex]
[tex]\[ x_2 = \frac{-(-1) - \sqrt{361}}{2 \cdot 1} \][/tex]
[tex]\[ x_2 = \frac{1 - 19}{2} \][/tex]
[tex]\[ x_2 = \frac{-18}{2} \][/tex]
[tex]\[ x_2 = -9 \][/tex]
Thus, the solutions to the equation [tex]\(x^2 - x - 90 = 0\)[/tex] are [tex]\(x_1 = 10\)[/tex] and [tex]\(x_2 = -9\)[/tex].
Finally, to find [tex]\(a + b\)[/tex], we add the solutions together:
[tex]\[ a + b = 10 + (-9) \][/tex]
[tex]\[ a + b = 1 \][/tex]
Therefore, the sum [tex]\(a + b\)[/tex] is:
[tex]\[ \boxed{1} \][/tex]
