The Quadratic Formula

To master the Quadratic Formula, you must first be an expert at identifying the Standard Form of a quadratic equation. Every quadratic equation on the SAT can be rewritten as:

$ax^2 + bx + c = 0$

Where $a$, $b$, and $c$ are constants, and $a \neq 0$.

The Formula

The Quadratic Formula provides the solutions (also called roots, x-intercepts, or zeros) for $x$:

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

Crucial Note: This formula is NOT provided on the SAT Reference Sheet. You must memorize it perfectly. A single sign error (forgetting the negative in $-b$ or the subtraction in $b^2 - 4ac$) will lead you directly to one of the "distractor" answer choices the SAT provides.

Identifying the Coefficients

The SAT will often try to trick you by presenting the equation out of order or with missing terms.

• If you see $3x^2 = 5x - 2$, you must subtract $5x$ and add $2$ to both sides to get $3x^2 - 5x + 2 = 0$. Here, $a=3, b=-5, c=2$.
• If a term is missing, such as $x^2 - 9 = 0$, then $b=0$.
• If there is no visible number in front of $x^2$, then $a=1$.

The Discriminant: The SAT’s Favorite Shortcut

The expression under the radical, $b^2 - 4ac$, is called the discriminant ($D$). On the SAT, you are often asked how many solutions an equation has rather than what the solutions are.

1. If $b^2 - 4ac > 0$: The equation has 2 distinct real solutions. (The graph crosses the x-axis twice).
2. If $b^2 - 4ac = 0$: The equation has 1 real solution (a "double root"). (The vertex of the parabola touches the x-axis).
3. If $b^2 - 4ac < 0$: The equation has 0 real solutions (but 2 complex/imaginary solutions). (The graph never touches the x-axis).

Sum and Product of Roots

While you can find the individual roots and add them, the SAT often asks for the "sum of the solutions." You can save significant time using these shortcuts derived from the formula:

• Sum of roots: $\frac{-b}{a}$
• Product of roots: $\frac{c}{a}$

The Geometry of the Formula

The first part of the formula, $\frac{-b}{2a}$, is the formula for the x-coordinate of the vertex of the parabola. The $\pm \frac{\sqrt{b^2-4ac}}{2a}$ part represents the distance you move to the left and right from the vertex to find the x-intercepts. Understanding this connection helps you visualize the problem on the coordinate plane.

1. The "Exact Value" Trigger

If you look at the answer choices and see square root symbols (e.g., $\frac{5 \pm \sqrt{13}}{2}$), do not waste time trying to factor. This is a clear signal that the Quadratic Formula is required.

2. Desmos vs. Manual Calculation

On the Digital SAT, the Desmos graphing calculator is always available.

• Use Desmos when: You need to find the number of solutions (just graph it and count the x-intercepts) or when the coefficients are large decimals.
• Use the Formula when: The answer choices are in radical form. Desmos will give you a decimal (like $2.414$), but the answer choices will look like $1 + \sqrt{2}$. You need the formula to match the exact radical form.

3. The "No Real Solution" Trap

When a question mentions "no real solutions" or "exactly one solution" and includes a constant like $k$ (e.g., $x^2 + kx + 9 = 0$), immediately write down the discriminant formula $b^2 - 4ac$. Set it $< 0$ for no solutions or $= 0$ for one solution, and solve for $k$.

4. Simplify Before You Start

If you have an equation like $10x^2 - 20x - 30 = 0$, divide the entire equation by $10$ first to get $x^2 - 2x - 3 = 0$. Smaller numbers lead to fewer calculation errors in the formula.