Question

find all zeros of (f(x) = x^3 + 4x^2 + x + 4). enter the zeros separated by commas.

(-4, i, -i)

question 11
find all zeros for the function: (g(m) = m^2 + 7m + 1). fully simplify all answers, including complex zeros.

(m =)

Explanation:

⚡ Using what you learned: quadratic formula and its applications

Step 1: Set the function to zero

To find the zeros of \( g(m) = m^2 + 7m + 1 \), set \( g(m) = 0 \):

$$ m^2 + 7m + 1 = 0 $$

Step 2: Apply the quadratic formula

Identify the coefficients:

$$ a = 1, \quad b = 7, \quad c = 1 $$

Substitute these values into the quadratic formula \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

$$ m = \frac{-7 \pm \sqrt{7^2 - 4(1)(1)}}{2(1)} $$

$$ m = \frac{-7 \pm \sqrt{49 - 4}}{2} $$

$$ m = \frac{-7 \pm \sqrt{45}}{2} $$

Step 3: Simplify the radical

Simplify \(\sqrt{45}\):

$$ \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5} $$

Substitute back into the expression:

$$ m = \frac{-7 \pm 3\sqrt{5}}{2} $$

Answer:

$$ \frac{-7 + 3\sqrt{5}}{2}, \frac{-7 - 3\sqrt{5}}{2} $$