How to factor quadratics on the ACT

To factor a quadratic like x^2 + bx + c, find two numbers that multiply to c and add to b. Before you start, always check for a common factor, and always check whether the expression is a difference of squares. Factoring is a pattern-recognition skill, so it gets fast surprisingly quickly once you have seen enough of them.

What factoring buys you

Factoring rewrites a sum as a product, and a product is powerful because of the zero product property: if two factors multiply to zero, one of them is zero. That turns a quadratic equation into two tiny linear ones.

Take x^2 - x - 12 = 0. It factors to (x - 4)(x + 3) = 0, so x = 4 or x = -3. Those two values are the roots of the equation and also the x-intercepts of the parabola y = x^2 - x - 12, meaning the graph crosses the x-axis at (4, 0) and (-3, 0). Roots, solutions, zeros, and x-intercepts are four names for the same thing, and the ACT will use all of them.

When the leading coefficient is 1

This is the workhorse case. You want two numbers whose product is the constant term and whose sum is the middle coefficient.

Factor x^2 + 7x + 12. Which pairs multiply to 12? Try 1 and 12, 2 and 6, 3 and 4. Only 3 and 4 add to 7, so the answer is (x + 3)(x + 4). Multiply it back out to confirm: x^2 + 4x + 3x + 12, which is x^2 + 7x + 12.

Factor x^2 - 9x + 20. You need a product of 20 and a sum of -9, which means both numbers are negative: -4 and -5. The answer is (x - 4)(x - 5).

Getting the signs right

Signs cause more lost points here than arithmetic does, but they follow a rule you can state in one breath. Look at the constant term first.

• Constant positive: both numbers share a sign, and the middle term tells you which. Positive middle means both positive; negative middle means both negative.
• Constant negative: the numbers have opposite signs, and the larger one in size takes the sign of the middle term.

Factor x^2 - 2x - 15. The constant is negative, so one number is positive and one is negative, and they must be 5 and 3 in size since those multiply to 15. The middle term is negative, so the bigger number carries the minus: -5 and +3. The answer is (x - 5)(x + 3). Sure enough, -5 times 3 is -15 and -5 plus 3 is -2.

Difference of squares and perfect squares

Two patterns are worth recognizing on sight, since they let you skip the search.

• Difference of squares: x^2 - 49 = (x - 7)(x + 7), and 9x^2 - 16 = (3x - 4)(3x + 4). Any square minus another square splits this way. Note there is no middle term.
• Perfect square trinomial: x^2 + 10x + 25 = (x + 5)^2. The first and last terms are squares, and the middle term is twice the product of their roots.

A sum of squares such as x^2 + 49 does not factor over the real numbers.

Pull out a common factor first

Before anything else, ask whether every term shares a factor. Removing it makes the rest easy and is often required for a fully factored answer.

Factor 2x^2 - 12x + 16. Every term is divisible by 2, so pull it out: 2(x^2 - 6x + 8). Now factor the inside with numbers that multiply to 8 and add to -6, which are -2 and -4. The full answer is 2(x - 2)(x - 4).

Sometimes the common factor includes a variable. For 3x^2 - 12x, pull out 3x to get 3x(x - 4), which means the roots are x = 0 and x = 4. Students lose the x = 0 solution constantly by dividing both sides by x instead of factoring it out. Never divide away a variable; you delete a solution.

When the leading coefficient is not 1

These appear less often, but they are very doable with grouping. Factor 3x^2 + 10x + 8. Multiply a times c, which is 3 times 8, or 24. Now find two numbers that multiply to 24 and add to 10: those are 6 and 4. Split the middle term using them:

• 3x^2 + 6x + 4x + 8
• Group and factor each pair: 3x(x + 2) + 4(x + 2)
• The shared (x + 2) comes out: (3x + 4)(x + 2)

Multiply back to confirm: 3x^2 + 6x + 4x + 8, which is 3x^2 + 10x + 8. If the grouping ever feels slower than the alternative, remember the quadratic formula is always available; see how to solve quadratic equations.

Traps and a quick drill

• Choosing the right pair of numbers but the wrong signs. Always add your two numbers mentally to confirm they give the middle term.
• Factoring an expression that is not yet set equal to zero and then treating the factors as solutions.
• Reading the roots off with the wrong sign. The factor (x + 3) gives the root x = -3, not 3.
• Missing the common factor and getting a partly factored answer.

Try these four:

• x^2 + 9x + 20
• x^2 - 5x - 24
• 4x^2 - 25
• 3x^2 - 27

Answers: (x + 4)(x + 5); (x - 8)(x + 3); (2x - 5)(2x + 5); and 3(x - 3)(x + 3). Ten minutes of this a day for a week and the pairs jump out at you, which is exactly what timed practice on thirty-six is for. For how much of the test this skill unlocks, see how much algebra is on the ACT.

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.