1. **State the problem:** Solve the quadratic equation $zx^2 + 10z - 24 = 0$ for $x$. 2. **Identify the coefficients:** Here, the quadratic is in terms of $x$, so $a = z$, $b = 0$, and $c = 10z - 24$ is incorrect because the equation is $zx^2 + 10z - 24 = 0$, which means $a = z$, $b = 0$, and $c = 10z - 24$ is not correct. Actually, the equation is $zx^2 + 10z - 24 = 0$, so the terms are $a = z$, $b = 10z$, and $c = -24$. 3. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 4. **Substitute the values:** $$x = \frac{-10z \pm \sqrt{(10z)^2 - 4 \cdot z \cdot (-24)}}{2z}$$ 5. **Simplify under the square root:** $$x = \frac{-10z \pm \sqrt{100z^2 + 96z}}{2z}$$ 6. **Factor inside the square root if possible:** $$\sqrt{100z^2 + 96z} = \sqrt{4z(25z + 24)} = 2\sqrt{z(25z + 24)}$$ 7. **Rewrite the expression:** $$x = \frac{-10z \pm 2\sqrt{z(25z + 24)}}{2z}$$ 8. **Cancel common factor 2 in numerator and denominator:** $$x = \frac{\cancel{2}(-5z \pm \sqrt{z(25z + 24)})}{\cancel{2}z} = \frac{-5z \pm \sqrt{z(25z + 24)}}{z}$$ 9. **Split the fraction:** $$x = -5 \pm \frac{\sqrt{z(25z + 24)}}{z}$$ **Final answer:** $$x = -5 \pm \frac{\sqrt{z(25z + 24)}}{z}$$