1. **State the problem:** Solve the equation $$2x^2 - 28x = -58$$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$2x^2 - 28x + 58 = 0$$ 3. **Simplify the equation:** Divide every term by 2 to simplify: $$\frac{2x^2}{2} - \frac{28x}{2} + \frac{58}{2} = \cancel{\frac{2}{2}}x^2 - \cancel{\frac{28}{2}}14x + \cancel{\frac{58}{2}}29 = 0$$ 4. **Use the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions are given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $$a=1$$, $$b=-14$$, and $$c=29$$. 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-14)^2 - 4(1)(29) = 196 - 116 = 80$$ 6. **Find the roots:** $$x = \frac{-(-14) \pm \sqrt{80}}{2(1)} = \frac{14 \pm \sqrt{80}}{2}$$ 7. **Simplify the square root:** $$\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}$$ 8. **Write the final solutions:** $$x = \frac{14 \pm 4\sqrt{5}}{2} = \frac{14}{2} \pm \frac{4\sqrt{5}}{2} = 7 \pm 2\sqrt{5}$$ **Answer:** $$x = 7 + 2\sqrt{5}$$ or $$x = 7 - 2\sqrt{5}$$