Every quadratic equation on the ACT yields to one of three methods: taking square roots, factoring, or the quadratic formula. Get the equation equal to zero first, then choose. Knowing which method fits which problem is most of the skill, and it takes far less practice than students expect.
Start by setting the equation equal to zero
A quadratic has an x^2 term and nothing higher. Standard form is ax^2 + bx + c = 0, and getting there matters because of the zero product property: if two things multiply to zero, at least one of them must be zero. That property is what makes factoring work, and it only works against zero.
So x^2 = 3x + 10 is not ready yet. Move everything to one side to get x^2 - 3x - 10 = 0. Now it factors as (x - 5)(x + 2) = 0, which gives x = 5 or x = -2. Check the first one in the original: 25 equals 15 plus 10. Check the second: 4 equals -6 plus 10. Both work, and a quadratic usually does have two solutions.
Method 1: take square roots
When there is no plain x term, skip everything else and undo the square directly. Just remember that squaring destroys sign information, so you take both roots.
Solve 3x^2 = 48. Divide by 3 to get x^2 = 16, so x = 4 or x = -4. Both check: 3 times 16 is 48 either way.
Solve (x - 4)^2 = 25. Take the square root of both sides: x minus 4 equals plus or minus 5. That splits into x - 4 = 5, giving x = 9, and x - 4 = -5, giving x = -1. Check x = -1: negative 1 minus 4 is negative 5, and negative 5 squared is 25.
Method 2: factor
Factoring is the fastest method when the numbers are friendly, which on the ACT they often are. Solve x^2 - 5x + 6 = 0. You need two numbers that multiply to 6 and add to -5, which are -2 and -3, so the equation becomes (x - 2)(x - 3) = 0. Set each factor to zero: x = 2 or x = 3.
Spend about fifteen seconds looking for a factorization. If nothing clean appears, switch methods rather than forcing it. The full toolkit lives in how to factor quadratics.
Method 3: the quadratic formula
The formula always works, factorable or not. In words: x equals negative b, plus or minus the square root of the quantity b squared minus 4ac, all divided by 2a. Written inline, that is x = (-b plus or minus the square root of (b^2 - 4ac)) / (2a). The ACT does not hand you a formula sheet, so this one belongs in memory alongside the rest of the formulas to know for ACT Math.
Solve 2x^2 + 5x - 3 = 0. Here a = 2, b = 5, and c = -3.
- Compute b squared minus 4ac: 25 minus 4 times 2 times negative 3, which is 25 plus 24, or 49.
- The square root of 49 is 7.
- So x equals negative 5 plus or minus 7, all over 4.
- Using plus: 2 divided by 4, which is 1/2. Using minus: negative 12 divided by 4, which is -3.
- Check x = 1/2: 2 times 1/4 is 1/2, plus 5/2 is 3, minus 3 is 0.
Two habits prevent most formula errors: write down a, b, and c explicitly before substituting, and put negative values in parentheses.
The discriminant tells you how many solutions exist
The piece under the square root, b^2 - 4ac, is called the discriminant, and sometimes a question asks only about it.
- Positive discriminant: two different real solutions.
- Zero: exactly one real solution, a repeated root.
- Negative: no real solutions.
For x^2 - 6x + 9 = 0, the discriminant is 36 minus 36, or 0, and indeed the equation is (x - 3)^2 = 0 with the single solution x = 3. For x^2 + 4x + 7 = 0, it is 16 minus 28, or -12, so there are no real solutions and the parabola never touches the x-axis.
Traps the ACT likes
- Forgetting the negative root. If x^2 = 36, the answer is x = 6 or x = -6, and a choice listing only 6 is bait.
- Factoring before the equation equals zero. Setting (x)(x - 3) = 10 to zero factor by factor produces nonsense.
- Sign errors in the discriminant when c is negative, since minus 4ac becomes addition.
- Solving correctly but answering the wrong prompt. Questions sometimes want the sum of the solutions, the positive solution only, or the larger one.
A quick self-check
Solve each with whichever method fits fastest:
- x^2 - 9 = 0
- x^2 + 7x + 10 = 0
- x^2 - 2x - 4 = 0
Answers: x = 3 or -3 by square roots; x = -2 or -5 by factoring; and for the third, the discriminant is 4 plus 16, or 20, so x equals 1 plus or minus the square root of 5. Practicing mixed sets on thirty-six trains the method choice itself, which is the part timed tests actually reward. If quadratics feel shaky, they show up on most lists of the hardest ACT Math topics for good reason, and they repay the practice.
Start practicing
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This article offers general ACT prep guidance. The ACT can change from year to year, including its format, scoring, policies, test dates, and fees, so always confirm the latest details on the official ACT website at act.org before you make decisions. ACT® is a registered trademark of ACT, Inc. thirty-six is not affiliated with, endorsed by, or sponsored by ACT.