Quadratic Equation 19E020

1. **State the problem:** Solve the quadratic equation $$x^2 + 5x + 6 = 0$$ using the quadratic formula and discriminant. 2. **Recall the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=1$$, $$b=5$$, and $$c=6$$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 5^2 - 4 \times 1 \times 6 = 25 - 24 = 1$$ 4. **Evaluate the square root of the discriminant:** $$\sqrt{\Delta} = \sqrt{1} = 1$$ 5. **Apply the quadratic formula:** $$x = \frac{-5 \pm 1}{2 \times 1} = \frac{-5 \pm 1}{2}$$ 6. **Find the two solutions:** - For the plus sign: $$x = \frac{-5 + 1}{2} = \frac{-4}{2} = -2$$ - For the minus sign: $$x = \frac{-5 - 1}{2} = \frac{-6}{2} = -3$$ 7. **Final answer:** The solutions to the equation $$x^2 + 5x + 6 = 0$$ are $$x = -2$$ and $$x = -3$$.