To solve this problem, we need to correctly substitute the coefficients of the quadratic equation [tex]\( 0 = x^2 - 9x - 20 \)[/tex] into the quadratic formula. The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
In the equation [tex]\( 0 = x^2 - 9x - 20 \)[/tex]:
- The coefficient [tex]\( a \)[/tex] is [tex]\( 1 \)[/tex].
- The coefficient [tex]\( b \)[/tex] is [tex]\( -9 \)[/tex].
- The constant term [tex]\( c \)[/tex] is [tex]\( -20 \)[/tex].
Let's substitute these values into the quadratic formula step-by-step:
1. Identify coefficients:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -9 \)[/tex]
- [tex]\( c = -20 \)[/tex]
2. Substitute [tex]\( b \)[/tex] into the formula:
[tex]\[ x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(-20)}}{2(1)} \][/tex]
Simplify the negative sign:
[tex]\[ x = \frac{9 \pm \sqrt{(-9)^2 - 4(1)(-20)}}{2(1)} \][/tex]
3. Calculate the discriminant part:
- Firstly, [tex]\( b^2 \)[/tex]:
[tex]\[ (-9)^2 = 81 \][/tex]
- Then, the term [tex]\( -4ac \)[/tex]:
[tex]\[ -4 \cdot 1 \cdot -20 = 80 \][/tex]
Combine these to form the discriminant:
[tex]\[ (-9)^2 - 4(1)(-20) = 81 - (-80) = 81 + 80 = 161 \][/tex]
4. Substitute the discriminant back:
[tex]\[ x = \frac{9 \pm \sqrt{161}}{2(1)} \][/tex]
So, we look for the equation that correctly represents these substitutions:
[tex]\[ x=\frac{-9 \pm \sqrt{(-9)^2-4(1)(-20)}}{2(1)} \][/tex]
Thus, the correct equation with the proper substitution is:
[tex]\[ x = \frac{-9 \pm \sqrt{(-9)^2 - 4(1)(-20)}}{2(1)} \][/tex]
Therefore, the first option is correct:
[tex]\[ \boxed{x=\frac{-9 \pm \sqrt{(-9)^2-4(1)(-20)}}{2(1)}} \][/tex]
