Quadratic Equations and Factoring on the ACT: Every Method, Named and Explained

Step 1: Set the quadratic equal to zero. Step 2: Identify b and c from the standard form x² + bx + c = 0. Step 3: List factor pairs of c — pairs of integers that multiply to c. Step 4: Find the pair that adds to b. Step 5: Write the factored form (x + p)(x + q) = 0. Step 6: Apply the Zero-Factor Rule (Concept 2) to find the roots.

Sign guidance: if c is positive, p and q have the same sign (both positive if b is positive; both negative if b is negative). If c is negative, p and q have opposite signs (the larger absolute value matches the sign of b).

After factoring: for each factor (x + p) or (x − p), set it equal to zero and solve. Never read roots directly from the parentheses. The three-second check: the factored form (x − r)(x − s) = 0 has roots x = r and x = s — both signs flipped from what is written. Write out the step explicitly every time, even when it feels obvious.

ACT application: when the question asks for the sum or product of the roots, use these identities for ax² + bx + c = 0: sum of roots = −b/a, product of roots = c/a. These shortcuts let you find the sum or product without factoring at all.

Step 1: Set the quadratic equal to zero in the form ax² + bx + c = 0. Step 2: Compute the product a × c. Step 3: Find two integers that multiply to ac and add to b. Call them m and n. Step 4: Rewrite the middle term bx as mx + nx. Step 5: Factor by grouping — group the first two terms and last two terms, factor out the GCF from each group. Step 6: Factor out the common binomial. Step 7: Apply the Zero-Factor Rule.

Alternative: if the quadratic is factorable and the answer choices are given, back-solving by plugging each answer choice into the original equation is often faster than the AC Method on the ACT.

Difference of squares: a² − b² = (a + b)(a − b). Recognize it when you see two perfect squares separated by a minus sign and no middle term. Examples: x² − 9 = (x+3)(x−3), 4x² − 25 = (2x+5)(2x−5), x² − 16 = (x+4)(x−4).

Perfect square trinomial: a² + 2ab + b² = (a + b)² or a² − 2ab + b² = (a − b)². Recognize it when the first and last terms are perfect squares and the middle term is twice the product of their square roots. Examples: x² + 6x + 9 = (x+3)², x² − 10x + 25 = (x−5)².

ACT application: difference of squares appears frequently in disguised forms where the variable has a coefficient, e.g., 9x² − 16. Always check for these patterns before starting the AC Method.

x = (−b ± √(b² − 4ac)) / (2a)

The expression under the radical, b² − 4ac, is called the discriminant. If the discriminant is positive, there are two real roots. If it is zero, there is exactly one repeated real root. If it is negative, there are no real roots (which the ACT occasionally asks about). Always identify a, b, and c from the standard form before substituting.

ACT shortcut: if the question asks how many real solutions the quadratic has without asking for their values, compute only the discriminant. There is no need to complete the full formula.

For the parabola y = a(x − r)(x − s) where r and s are the roots:

X-intercepts: (r, 0) and (s, 0). These are the points where y = 0, which are exactly the roots you find by factoring.

Axis of symmetry: x = (r + s)/2. The axis of symmetry is always the average of the two roots. For y = ax² + bx + c, this equals x = −b/(2a). Both formulas give the same answer.

Vertex x-coordinate: same as the axis of symmetry, x = (r + s)/2. The vertex is the maximum or minimum point of the parabola, and its x-coordinate sits exactly halfway between the two roots.

Direction: if a > 0, the parabola opens upward (U-shape); if a < 0, it opens downward (arch-shape). This is determined solely by the sign of the leading coefficient.

Area/geometry word problems: when length and width are both expressed in terms of the same variable and a product (area) is given, expand and set equal to zero. The resulting equation is almost always a quadratic.

Equation not equal to zero: if you see ax² + bx + c = k where k ≠ 0, subtract k from both sides first to get ax² + bx + (c − k) = 0, then factor or apply the formula. Never factor a quadratic that is not set equal to zero.

Function notation: if f(x) = x² + bx + c and the question asks for values of x where f(x) = k, substitute k and rearrange to standard form before factoring.

Projectile/height problems: height = −16t² + v⊂0;t + h⊂0; (in feet) or h = −4.9t² + v⊂0;t + h⊂0; (in meters). Setting height equal to zero finds when the object hits the ground; setting height equal to a given value finds when it reaches that height.