ACT Math: Solve Quadratics Using Factoring, Completing the Square, and the Quadratic Formula

Factoring: x^2+5x+6=0 factors to (x+2)(x+3)=0, so x=-2 or -3. Fastest when the quadratic factors nicely. Completing the square: x^2+4x=0 becomes (x+2)^2-4=0, so (x+2)^2=4, x+2=±2, x=0 or -4. Useful when you need the vertex form or when factoring is messy. Quadratic formula: x=(-b±sqrt(b^2-4ac))/(2a) works for any quadratic ax^2+bx+c=0. Always reliable, even when factoring is hard. Choosing the right method saves time; if a quadratic factors nicely, factoring is fastest. If not, the quadratic formula is reliable.

Example: 2x^2+3x+1=0. Factoring: (2x+1)(x+1)=0, x=-1/2 or -1 (fast). Example: x^2+2x+5=0. Discriminant b^2-4ac=4-20=-16<0, so no real solutions. Using the quadratic formula instantly reveals this; factoring would be frustrating.

Trap 1: Using the quadratic formula when factoring is faster. If x^2-5x+6=0, factoring (x-2)(x-3)=0 is much faster than the formula. Train yourself to scan for factors first; if they don't jump out in 5 seconds, use the formula. Trap 2: Forgetting the ± in the quadratic formula or completing the square. (x+2)^2=9 gives x+2=±3, so x=1 or -5. Missing the ±results in losing one solution. After solving with any method, verify by plugging your solutions back into the original equation. This check catches errors instantly.

When you see a quadratic, spend 5 seconds looking for factors. If none appear, use the quadratic formula without hesitation. Don't waste time struggling with factoring when the formula will work reliably.

Problem 1: x^2-7x+12=0. Factoring is fastest: (x-3)(x-4)=0, x=3 or 4. Problem 2: x^2+x-1=0. Doesn't factor nicely. Use formula: x=(-1±sqrt(1+4))/2=(-1±sqrt(5))/2. Problem 3: 2x^2-8=0. Solve directly: x^2=4, x=±2 (no factoring or formula needed). Problem 4: x^2+4x+4=0. Recognize perfect square: (x+2)^2=0, x=-2 (double root). Problem 5: 3x^2+6x+3=0. Factor out 3: 3(x^2+2x+1)=0, 3(x+1)^2=0, x=-1. Problem 6: x^2-3x+5=0. Discriminant is negative (9-20=-11<0), so no real solutions. Each problem uses a different approach; recognizing which is fastest is the key skill.

Do ten more quadratic problems daily, choosing the fastest method for each. By test day, you'll assess each quadratic and solve it in seconds.

Quadratic equations appear throughout ACT Math (algebra, coordinate geometry, word problems). Mastering all three solution methods ensures you can solve any quadratic quickly and correctly. Students who know all three methods solve quadratics faster than peers and gain time for harder problems.

This week, practice all three methods. Learn to recognize which is fastest for each quadratic. By test day, solving quadratics will feel automatic and you'll use the optimal method instantly.