ACT Math: Use the Quadratic Formula When Factoring Fails

Formula: x=(-b±√(b²-4ac))/(2a). For the equation ax²+bx+c=0, identify a, b, and c, then plug into the formula. Example: 2x²+5x+3=0. Here a=2, b=5, c=3. x=(-5±√(25-24))/4=(-5±1)/4. So x=-1 or x=-1.5. The quadratic formula is guaranteed to work, even when factoring is impossible or difficult.

The ± symbol means there are (usually) two solutions. The expression under the square root (b²-4ac) is called the discriminant. If it's positive, two real solutions exist. If zero, one solution. If negative, no real solutions (complex solutions exist, but ACT doesn't test these).

Use factoring if the quadratic factors easily (small, nice numbers). Example: x²+5x+6=0 factors as (x+2)(x+3)=0, so x=-2 or x=-3. Fast. Use the quadratic formula if factoring looks difficult or impossible. Example: 2x²+7x+1=0 doesn't factor nicely; use the formula. On ACT, if the numbers are messy or non-integer, the quadratic formula is usually the intended method.

Strategy: Before using the formula, check if the quadratic factors. If the discriminant is a perfect square, factoring might work. If it's not a perfect square, use the formula to avoid messy factoring attempts.

Mistake 1: Forgetting the ± symbol (reporting only one solution when two exist). Fix: Always acknowledge both ± solutions unless the question asks for a specific one. Mistake 2: Mixing up signs (using +b instead of -b, or forgetting -4ac in the discriminant). Fix: Write out a, b, c values before plugging in. Mistake 3: Arithmetic errors in the discriminant calculation. Fix: Compute b²-4ac slowly and double-check. Mistake 4: Forgetting to divide by 2a at the end. Fix: The entire numerator (-b±√...) is divided by 2a, not just the square root. These four mistakes cause 95% of quadratic formula errors.

Drill: Solve using the quadratic formula. (1) x²-4x+3=0. (a=1, b=-4, c=3). x=(4±√(16-12))/2=(4±2)/2. x=3 or x=1. (2) 2x²-3x-2=0. (a=2, b=-3, c=-2). x=(3±√(9+16))/4=(3±5)/4. x=2 or x=-0.5.

The quadratic formula is mechanical and guaranteed to work. Even if you can't factor, you can still solve any quadratic equation on ACT. Knowing the formula gives you confidence that you can solve any quadratic, regardless of complexity.

Memorize the formula this week (many students write it on their scratch paper at the start of the test for peace of mind). Solve fifteen quadratic equations using the formula, checking your work by substituting solutions back into the original equation. By test day, applying the formula will feel automatic, and you'll solve quadratics reliably under time pressure.