The solutions to a quadratic equation of the form:

$ax^2 + bx + c = 0$

are given by the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Explanation:

1. Discriminant $\Delta = b^2 - 4ac$:

   • If $\Delta > 0$, there are two distinct real solutions.
   • If $\Delta = 0$, there is one real solution (a repeated root).
   • If $\Delta < 0$, there are two complex conjugate solutions.

2. Factoring (if possible):

   • If the quadratic can be factored as $(x - r)(x - s) = 0$, then the solutions are simply $x = r$ and $x = s$.

3. Completing the Square:

   • Another method to solve quadratics is by rewriting it in the form $(x + d)^2 = e$ and solving for $x$.

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Related Questions:

1. How do you derive the quadratic formula?
2. How can you determine the nature of roots without solving the equation?
3. What are real-life applications of quadratic equations?
4. How do you solve a quadratic equation graphically?
5. How do you use the quadratic formula for complex numbers?

Tip:

Always check the discriminant first—it tells you what kind of solutions to expect!