Question

question 16 (5 points)
write \\(-8 + \sqrt{-12}\\) as a complex number.
a) \\(-8 - 12i\\)
b) \\(-8 + 2\sqrt{3}i\\)
c) \\(-8 + 4\sqrt{3}i\\)
d) \\(-8 + 12i\\)

question 17 (5 points)
solve \\(x^2 + 13x + 22 = 0\\) using the quadratic formula.
a) \\(x = 2, 11\\)
b) \\(x = 4, 22\\)
c) \\(x = -4, -22\\)
d) \\(x = -2, -11\\)

Explanation:

Simplify the radical with a negative radicand

Using the Simplifying Radicals with Negative Radicands knowledge point

$$ \sqrt{-12} = \sqrt{-1 \cdot 4 \cdot 3} = \sqrt{-1} \cdot \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}i $$

Write as a complex number

Using the Complex Number Representation knowledge point

$$ -8 + \sqrt{-12} = -8 + 2\sqrt{3}i $$

Identify quadratic coefficients

We are given the quadratic equation:

$$ x^2 + 13x + 22 = 0 $$

The coefficients are:

$$ a = 1, \quad b = 13, \quad c = 22 $$

Apply the quadratic formula

Using the Quadratic Formula knowledge point

$$
LATEXBLOCK0
$$

Calculate the solutions

Using the Quadratic Formula knowledge point

$$
LATEXBLOCK1
$$

Thus, the solutions are \(x = -2, -11\).

Answer:

Question 16

• A) \(-8 - 12i\)
• B) \(-8 + 2\sqrt{3}i\) (Correct answer)
• C) \(-8 + 4\sqrt{3}i\)
• D) \(-8 + 12i\)

Question 17

• A) \(x = 2, 11\)
• B) \(x = 4, 22\)
• C) \(x = -4, -22\)
• D) \(x = -2, -11\) (Correct answer)