1. **State the problem:** Simplify the radicand expressions and solve the quadratic equation $-x^2 + 7x - 19 = 0$ using the quadratic formula. 2. **Simplify each radicand:** - For $\sqrt{-49}$, rewrite as $\sqrt{-1 \times 49} = \sqrt{-1} \times \sqrt{49} = i \times 7 = 7i$. Since square roots can be positive or negative, the answer is $\pm 7i$. - For $\sqrt{64}$, since 64 is positive and a perfect square, $\sqrt{64} = 8$. The answer is $\pm 8$. - For $\sqrt{-9}$, rewrite as $\sqrt{-1 \times 9} = i \times 3 = 3i$. So, $\pm 3i$. - For $\sqrt{-24}$, rewrite as $\sqrt{-1 \times 24} = i \sqrt{24}$. Simplify $\sqrt{24} = \sqrt{4 \times 6} = 2 \sqrt{6}$. So, $\pm 2i \sqrt{6}$. - For $\sqrt{-6}$, rewrite as $\sqrt{-1 \times 6} = i \sqrt{6}$. So, $\pm i \sqrt{6}$. 3. **Solve the quadratic equation $-x^2 + 7x - 19 = 0$ using the quadratic formula:** The quadratic formula is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Given $a = -1$, $b = 7$, and $c = -19$. Calculate the discriminant: $$b^2 - 4ac = 7^2 - 4(-1)(-19) = 49 - 76 = -27$$ Since the discriminant is negative, the solutions are complex. Substitute into the formula: $$x = \frac{-7 \pm \sqrt{-27}}{2(-1)} = \frac{-7 \pm \sqrt{-1 \times 27}}{-2} = \frac{-7 \pm i \sqrt{27}}{-2}$$ Simplify $\sqrt{27}$: $$\sqrt{27} = \sqrt{9 \times 3} = 3 \sqrt{3}$$ So, $$x = \frac{-7 \pm 3i \sqrt{3}}{-2}$$ Divide numerator and denominator by $-1$: $$x = \frac{7 \mp 3i \sqrt{3}}{2}$$ 4. **Final answers:** - $\sqrt{-49} = \pm 7i$ - $\sqrt{64} = \pm 8$ - $\sqrt{-9} = \pm 3i$ - $\sqrt{-24} = \pm 2i \sqrt{6}$ - $\sqrt{-6} = \pm i \sqrt{6}$ - Solutions to $-x^2 + 7x - 19 = 0$ are: $$x = \frac{7 \pm 3i \sqrt{3}}{2}$$