Quadratic Equations

A quadratic equation is one that can be written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are numbers and $a \neq 0$. Examples: $x^2 - 7x + 10 = 0$, $2x^2 + 5x - 3 = 0$, $x^2 = 16$.

There are three ways to solve a quadratic on the SAT, and you should know all three:

1. Factoring (fastest when it works). Find two numbers that multiply to $c$ and add to $b$. Rewrite the equation as $(x - p)(x - q) = 0$, then set each factor to zero.

For $x^2 - 7x + 10 = 0$: find numbers that multiply to $10$ and add to $-7$. Those are $-2$ and $-5$. So $(x - 2)(x - 5) = 0$, giving solutions $x = 2$ and $x = 5$.

2. The quadratic formula (always works).

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

The expression under the square root, $b^2 - 4ac$, is the discriminant. It tells you how many real solutions exist:

• Positive: two distinct solutions.
• Zero: one repeated solution.
• Negative: no real solutions (the parabola never crosses zero).

3. Square-rooting both sides (special cases). If the equation is just $x^2 = 16$, take the square root of both sides: $x = \pm 4$. Don't forget the negative root — that trips students often.

A few related ideas:

• The graph of a quadratic is a parabola — a U-shape that opens up if $a > 0$, down if $a < 0$.
• The x-intercepts of the parabola are the solutions to the equation when set to zero.
• The vertex (highest or lowest point) is at $x = -b / (2a)$.