Let's go through each equation one by one and solve for [tex]\( u \)[/tex]:
### Equation 1:
[tex]\[ u^2 - 17u - 16 = 0 \][/tex]
To solve this quadratic equation, we can use the quadratic formula [tex]\( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -17 \)[/tex], and [tex]\( c = -16 \)[/tex].
1. Calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac = (-17)^2 - 4(1)(-16) = 289 + 64 = 353 \][/tex]
2. The roots are:
[tex]\[ u = \frac{17 \pm \sqrt{353}}{2} \][/tex]
So, the solutions for [tex]\( u \)[/tex] are:
[tex]\[ u = \frac{17 + \sqrt{353}}{2} \quad \text{and} \quad u = \frac{17 - \sqrt{353}}{2} \][/tex]
### Equation 2:
[tex]\[ 17u^2 + u + 16 = 0 \][/tex]
Again, use the quadratic formula [tex]\( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 17 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = 16 \)[/tex].
1. Calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac = (1)^2 - 4(17)(16) = 1 - 1088 = -1087 \][/tex]
Since the discriminant is negative, there are no real solutions. The roots are complex:
[tex]\[ u = \frac{-1 \pm \sqrt{-1087}}{34} = \frac{-1 \pm i\sqrt{1087}}{34} \][/tex]
So, the solutions for [tex]\( u \)[/tex] are:
[tex]\[ u = \frac{-1 + i\sqrt{1087}}{34} \quad \text{and} \quad u = \frac{-1 - i\sqrt{1087}}{34} \][/tex]
### Equation 3:
[tex]\[ -u^2 + 17u + 16 = 0 \][/tex]
To make it easier, we can multiply through by [tex]\(-1\)[/tex]:
[tex]\[ u^2 - 17u - 16 = 0 \][/tex]
We solved this in Equation 1, so the solutions are the same:
[tex]\[ u = \frac{17 + \sqrt{353}}{2} \quad \text{and} \quad u = \frac{17 - \sqrt{353}}{2} \][/tex]
### Equation 4:
[tex]\[ u^2 - 17u + 16 = 0 \][/tex]
Using the quadratic formula [tex]\( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -17 \)[/tex], and [tex]\( c = 16 \)[/tex]:
1. Calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac = (-17)^2 - 4(1)(16) = 289 - 64 = 225 \][/tex]
2. The roots are:
[tex]\[ u = \frac{17 \pm \sqrt{225}}{2} = \frac{17 \pm 15}{2} \][/tex]
So, the solutions for [tex]\( u \)[/tex] are:
[tex]\[ u = \frac{17 + 15}{2} = 16 \quad \text{and} \quad u = \frac{17 - 15}{2} = 1 \][/tex]
### Summary of Solutions
- Equation 1: [tex]\( u = \frac{17 + \sqrt{353}}{2} \)[/tex] and [tex]\( u = \frac{17 - \sqrt{353}}{2} \)[/tex]
- Equation 2: [tex]\( u = \frac{-1 + i\sqrt{1087}}{34} \)[/tex] and [tex]\( u = \frac{-1 - i\sqrt{1087}}{34} \)[/tex]
- Equation 3: [tex]\( u = \frac{17 + \sqrt{353}}{2} \)[/tex] and [tex]\( u = \frac{17 - \sqrt{353}}{2} \)[/tex]
- Equation 4: [tex]\( u = 16 \)[/tex] and [tex]\( u = 1 \)[/tex]
These are the solutions for the equations written in terms of [tex]\( u \)[/tex].
