Quadratic Equation Use F152F7

1. Let's start by understanding what a quadratic equation is. A quadratic equation is any equation that can be written in the form $$ax^2 + bx + c = 0$$ where $a$, $b$, and $c$ are constants and $a \neq 0$. 2. Now, why is this important in daily life or AI/ML? Quadratic equations help us model situations where the relationship between variables is not just straight lines but curves, like parabolas. For example, in physics, the path of a thrown ball follows a quadratic equation. 3. In AI and machine learning, quadratic equations appear in optimization problems. For instance, when training models, we often minimize a quadratic cost function to find the best parameters. 4. The formula to solve quadratic equations is called the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ This formula gives the solutions (roots) of the quadratic equation. 5. Important rules: - The term under the square root, $b^2 - 4ac$, is called the discriminant. - If the discriminant is positive, there are two real solutions. - If it is zero, there is one real solution. - If it is negative, the solutions are complex (not real). 6. Let's see an example: Suppose you want to find when a ball thrown upwards will hit the ground. The height $h$ at time $t$ might be modeled as: $$h = -5t^2 + 20t + 15$$ To find when it hits the ground, set $h=0$: $$-5t^2 + 20t + 15 = 0$$ 7. Divide both sides by $-5$ to simplify: $$\cancel{-5}t^2 + \cancel{-5} \times (-4)t + \cancel{-5} \times (-3) = 0 \Rightarrow t^2 - 4t - 3 = 0$$ 8. Apply the quadratic formula with $a=1$, $b=-4$, $c=-3$: $$t = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \times 1 \times (-3)}}{2 \times 1} = \frac{4 \pm \sqrt{16 + 12}}{2} = \frac{4 \pm \sqrt{28}}{2}$$ 9. Simplify the square root: $$\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}$$ So, $$t = \frac{4 \pm 2\sqrt{7}}{2} = 2 \pm \sqrt{7}$$ 10. The two solutions are approximately: $$t = 2 + 2.65 = 4.65$$ seconds and $$t = 2 - 2.65 = -0.65$$ seconds (which we discard since time cannot be negative). 11. So, the ball hits the ground after about 4.65 seconds. 12. In summary, quadratic equations help us solve real-world problems involving curved relationships, and in AI/ML, they are fundamental in optimization and modeling nonlinear data.