1. **State the problem:** Solve the quadratic equation $$x^2 - 14 = 5x$$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$x^2 - 5x - 14 = 0$$ 3. **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, $a=1$, $b=-5$, and $c=-14$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 1 \times (-14) = 25 + 56 = 81$$ 5. **Find the roots:** $$x = \frac{-(-5) \pm \sqrt{81}}{2 \times 1} = \frac{5 \pm 9}{2}$$ 6. **Evaluate each root:** - For the plus sign: $$x = \frac{5 + 9}{2} = \frac{14}{2} = 7$$ - For the minus sign: $$x = \frac{5 - 9}{2} = \frac{-4}{2} = -2$$ 7. **Final answer:** The solutions to the equation are: $$x = \{-2, 7\}$$ These are the values of $x$ that satisfy the original equation.