Question

1. in order to simplify i^n where n is...

a. true
b. false

1. identify the correct solutions below. \\(x^2 - x - 30\\)

a. \\((-5, 6)\\)
b. \\((-\infty, -5) \cup (6, \infty)\\)
c. \\((-\infty, -6) \cup (5, \infty)\\)
d. \\((-6, 5)\\)

1. solve using the quadratic formula. \\(x^2 - 4x + 5\\)

a. \\(x = 2 + i, 2 - i\\)
b. \\(x = -5, 1\\)
c. \\(x = 5, -1\\)
d. \\(x = -2 + i, -2 - i\\)

1. solve by taking the square root: \\(-3x^2 = 27\\)

a. \\(x = \pm 3i\\)
b. \\(x = \pm 9i\\)
c. \\(x = \pm 3\\)
d. \\(x = \pm \sqrt{30}\\)

Explanation:

Step1: Solve Q10: Factor the quadratic

The quadratic is $x^2 - x - 30 = 0$. Factor:
$(x-6)(x+5)=0$
Roots: $x=6, x=-5$. Assuming it's $x^2 - x - 30 > 0$, the solution is $(-\infty,-5)\cup(6,\infty)$.

Step2: Solve Q11: Apply quadratic formula

For $x^2 - 4x + 5 = 0$, quadratic formula is $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.
Here $a=1, b=-4, c=5$.
$\Delta=(-4)^2-4(1)(5)=16-20=-4$
$x=\frac{4\pm\sqrt{-4}}{2}=\frac{4\pm2i}{2}=2\pm i$

Step3: Solve Q12: Isolate $x^2$ and solve

Start with $-3x^2=27$. Divide by $-3$:
$x^2=-9$
Take square root: $x=\pm\sqrt{-9}=\pm3i$

Answer:

1. b. $(-\infty,-5)\cup(6,\infty)$
2. a. $x = 2 + i,\ 2 - i$
3. a. $x = \pm3i$