Tactics for Finding the Roots of Quadratic Equations
Master every method — then pick the fastest one on test day.
SAT Key Insight: A quadratic equation $ax^2 + bx + c = 0$ can have 0, 1, or 2 real roots. The SAT tests whether you can find them quickly and accurately.
1. Methods for Finding Roots
1A. Factoring Method
If you can rewrite $ax^2 + bx + c$ as a product of two linear factors, each factor set to zero gives a root.
$$ax^2 + bx + c = (px + q)(rx + s) = 0$$
$$\Rightarrow \quad x = -\frac{q}{p} \quad \text{or} \quad x = -\frac{s}{r}$$
Key Factoring Patterns
Difference of Squares
$$a^2 - b^2 = (a-b)(a+b)$$
Example: $x^2 - 25 = (x-5)(x+5)$
Perfect Square Trinomial
$$a^2 \pm 2ab + b^2 = (a \pm b)^2$$
Example: $x^2 + 6x + 9 = (x+3)^2$
AC Method
Find two numbers that multiply to $ac$ and add to $b$.
For $2x^2+7x+3$: $ac=6$, find $1 \times 6 = 6$ and $1+6=7$ ✓
1B. Square Root Method
When the equation has the form $x^2 = k$ or $(x - h)^2 = k$, take the square root of both sides. Don't forget ±!
$$(x - h)^2 = k \quad \Rightarrow \quad x - h = \pm\sqrt{k} \quad \Rightarrow \quad x = h \pm \sqrt{k}$$
1C. Completing the Square
Transform $ax^2 + bx + c = 0$ into the form $a(x-h)^2 = k$, then use the square root method.
Steps:
- Move the constant to the right: $ax^2 + bx = -c$
- If $a \neq 1$, divide everything by $a$: $x^2 + \frac{b}{a}x = -\frac{c}{a}$
- Add $\left(\frac{b}{2a}\right)^2$ to both sides
- Factor the left side as a perfect square
- Take $\pm\sqrt{\phantom{x}}$ of both sides and solve
1D. The Quadratic Formula
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
This ALWAYS works for any quadratic $ax^2 + bx + c = 0$.