Algebra - Solving quadratic equations by factoring and using the quadratic formula

A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a \neq 0$. Visually, the graph of a quadratic equation is a U-shaped curve called a parabola. If $a > 0$, the parabola opens upwards like a cup; if $a < 0$, it opens downwards like a cap.

The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. For example, if $(x - p)(x - q) = 0$, then $x - p = 0$ or $x - q = 0$. Visually, these values $p$ and $q$ represent the x-intercepts where the parabola crosses the horizontal x-axis.

Factoring Monic Trinomials involves solving $x^2 + bx + c = 0$ by finding two numbers that multiply to give $c$ and add to give $b$. Once found, the equation is written as $(x + m)(x + n) = 0$, allowing us to solve for $x$.

Factoring Non-Monic Trinomials (where $a \neq 1$) often requires 'splitting the middle term.' We look for two numbers that multiply to $a \times c$ and add to $b$. This allows the equation to be grouped and factored into two binomials.

The Quadratic Formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ is a universal method used to find the roots of any quadratic equation, especially when the expression cannot be easily factored into integers.

The Discriminant, denoted as $\Delta = b^2 - 4ac$, determines the nature of the roots. Visually: if $\Delta > 0$, the parabola intersects the x-axis at two distinct points; if $\Delta = 0$, the vertex of the parabola touches the x-axis at exactly one point; if $\Delta < 0$, the parabola never touches the x-axis, resulting in no real solutions.