To solve the quadratic equation [tex]\( x^2 - 8x - 9 = 0 \)[/tex], we need to find the roots of the equation. This can be done using the quadratic formula, which 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 equation [tex]\( ax^2 + bx + c = 0 \)[/tex].
For the given quadratic equation [tex]\( x^2 - 8x - 9 = 0 \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -8 \)[/tex]
- [tex]\( c = -9 \)[/tex]
Next, we calculate the discriminant, which is [tex]\( b^2 - 4ac \)[/tex]:
[tex]\[ \text{Discriminant} = (-8)^2 - 4 \cdot 1 \cdot (-9) = 64 + 36 = 100 \][/tex]
The discriminant is positive, which means the quadratic equation has two distinct real roots.
Now, we can find the roots using the quadratic formula:
[tex]\[ x_1 = \frac{-(-8) + \sqrt{100}}{2 \cdot 1} = \frac{8 + 10}{2} = \frac{18}{2} = 9 \][/tex]
[tex]\[ x_2 = \frac{-(-8) - \sqrt{100}}{2 \cdot 1} = \frac{8 - 10}{2} = \frac{-2}{2} = -1 \][/tex]
Therefore, the solutions to the quadratic equation [tex]\( x^2 - 8x - 9 = 0 \)[/tex] are:
[tex]\[ x = 9 \quad \text{and} \quad x = -1 \][/tex]
Among the given options, the correct solutions are:
- [tex]\( -1 \)[/tex]
- [tex]\( 9 \)[/tex]
