1. **State the problem:** Solve the quadratic equation $x^2 + 10x = 30$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$x^2 + 10x - 30 = 0$$ 3. **Identify coefficients:** For the quadratic equation $ax^2 + bx + c = 0$, here $a=1$, $b=10$, and $c=-30$. 4. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 10^2 - 4 \times 1 \times (-30) = 100 + 120 = 220$$ 6. **Find the square root of the discriminant:** $$\sqrt{220} = \sqrt{4 \times 55} = 2\sqrt{55}$$ 7. **Substitute values into the formula:** $$x = \frac{-10 \pm 2\sqrt{55}}{2}$$ 8. **Simplify the expression:** $$x = -5 \pm \sqrt{55}$$ 9. **Final solutions:** $$x_1 = -5 + \sqrt{55}$$ $$x_2 = -5 - \sqrt{55}$$ These are the two solutions to the quadratic equation.