1. **Problem:** Given one complex solution to the quadratic equation $f(x) = x^2 + 2x + 5$ is $w_1 = -1 + 2i$, find the other solution $w_2$ and state the relationship between $w_1$ and $w_2$. 2. **Formula and rules:** For a quadratic equation $ax^2 + bx + c = 0$, the sum and product of roots $w_1$ and $w_2$ satisfy: $$w_1 + w_2 = -\frac{b}{a}$$ $$w_1 w_2 = \frac{c}{a}$$ Since coefficients are real, complex roots occur in conjugate pairs. 3. **Find $w_2$:** Given $w_1 = -1 + 2i$, the other root is its complex conjugate: $$w_2 = -1 - 2i$$ 4. **Relationship:** $w_1$ and $w_2$ are complex conjugates. --- 5. **Problem:** Determine the modulus of $w_1 = -1 + 2i$. 6. **Formula:** The modulus of a complex number $a + bi$ is: $$|w| = \sqrt{a^2 + b^2}$$ 7. **Calculate modulus:** $$|w_1| = \sqrt{(-1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5}$$ **Final answers:** - The other solution is $w_2 = -1 - 2i$. - $w_1$ and $w_2$ are complex conjugates. - The modulus of $w_1$ is $\sqrt{5}$.