1. **Problem:** You want to know how to do a quadratic equation. 2. A quadratic equation is usually written as $$ax^2+bx+c=0$$ where $a\neq 0$. 3. The three most common ways to solve it are factoring, using the quadratic formula, or completing the square. 4. **Factoring rule:** If you can write the quadratic as $$ax^2+bx+c=(px+q)(rx+s)$$ then set each factor equal to $0$. 5. **Key zero-product rule:** If $$AB=0$$ then $A=0$ or $B=0$. 6. Example with factoring: solve $$x^2+5x+6=0$$. 7. Factor the expression: $$x^2+5x+6=(x+2)(x+3)$$. 8. Set each factor equal to $0$: $$x+2=0$$ or $$x+3=0$$. 9. Solve each one: $$x=-2$$ or $$x=-3$$. 10. **Quadratic formula rule:** For $$ax^2+bx+c=0$$, the solutions are $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$. 11. The part under the square root, $$b^2-4ac$$, is called the discriminant. 12. If the discriminant is positive, there are 2 real solutions. 13. If it is zero, there is 1 real solution. 14. If it is negative, there are no real solutions. 15. Example with the formula: solve $$2x^2+3x-2=0$$. 16. Identify the values: $a=2$, $b=3$, $c=-2$. 17. Substitute into the formula: $$x=\frac{-3\pm\sqrt{3^2-4(2)(-2)}}{2(2)}$$. 18. Simplify inside the square root: $$x=\frac{-3\pm\sqrt{9+16}}{4}$$. 19. Continue simplifying: $$x=\frac{-3\pm\sqrt{25}}{4}$$. 20. So $$x=\frac{-3\pm 5}{4}$$. 21. Split into two answers: $$x=\frac{-3+5}{4}$$ and $$x=\frac{-3-5}{4}$$. 22. Final answers: $$x=\frac{1}{2}$$ and $$x=-2$$. 23. **Completing the square rule:** Rewrite the quadratic so one side becomes a perfect square trinomial. 24. This method is useful when factoring is hard. 25. A simple plan is: move the constant, make the $x^2$ coefficient $1$ if needed, take half of $b$, square it, then add it to both sides. 26. **Best habit:** Always check your answer by substituting it back into the original equation. 27. If you want, I can also show you how to solve one specific quadratic step by step.