Factoring and Quadratics

Factoring is the distributive property run backward, and quadratics are what you can solve once you can factor. The test asks for both directions: multiply these out, and factor this.

Multiplying binomials: FOIL

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First, Outer, Inner, Last.

(x + 3)(x + 5)

• First: x times x is x squared.
• Outer: x times 5 is 5x.
• Inner: 3 times x is 3x.
• Last: 3 times 5 is 15.

Combine the middle: x squared + 8x + 15.

Notice what happened to the 3 and the 5. They multiplied to give the last term and added to give the middle coefficient. That is the observation the whole factoring method is built on.

With a negative:

(x - 4)(x + 6)

• First: x squared.
• Outer: 6x.
• Inner: -4x.
• Last: -24.

Result: x squared + 2x - 24. The signs travel with the numbers, so -4 and +6 multiply to -24 and add to +2.

Two patterns worth recognizing

Difference of squares: (a + b)(a - b) = a squared - b squared.

The middle terms cancel, which is why the result has only two terms.

• x squared - 49 factors as (x + 7)(x - 7).
• 4x squared - 25 factors as (2x + 5)(2x - 5).

Recognize it by two perfect squares with a minus between them. This is the single most recognizable pattern on the test.

A sum of squares does not factor. x squared + 49 is as simplified as it gets. A choice claiming to factor it is wrong by construction.

Perfect square trinomials:

• (a + b) squared = a squared + 2ab + b squared.
• (a - b) squared = a squared - 2ab + b squared.

So x squared + 10x + 25 is (x + 5) squared.

(a + b) squared is not a squared + b squared. The middle term is real and dropping it is the most common error in this topic.

Pulling out the greatest common factor

Do this first, always.

Factor 6x squared + 9x

Both terms share 3x: 3x(2x + 3).

Factor 4x squared - 36

Both share 4: 4(x squared - 9). Now the inside is a difference of squares, so the full factorization is 4(x + 3)(x - 3).

Stopping at 4(x squared - 9) is incomplete, and "factor completely" means keep going until nothing more comes out.

Factoring a trinomial

For x squared + bx + c, with a leading coefficient of 1:

Find two numbers that multiply to c and add to b.

Factor x squared + 11x + 24

Two numbers multiplying to 24 and adding to 11. Factor pairs of 24: 1 and 24, 2 and 12, 3 and 8, 4 and 6. The pair 3 and 8 adds to 11.

Answer: (x + 3)(x + 8).

Reading the signs

The signs of c and b tell you the signs of your two numbers before you start hunting.

Factor x squared - 7x + 12

c is positive and b is negative, so both numbers are negative. Pairs of 12 that add to -7: -3 and -4.

Answer: (x - 3)(x - 4).

Factor x squared + 2x - 15

c is negative, so the numbers have opposite signs. They multiply to -15 and add to +2: that is 5 and -3.

Answer: (x + 5)(x - 3).

When c is negative, the larger number takes the sign of b. Here b is positive, so the 5 is the positive one.

Solving a quadratic equation

Get zero on one side, factor, set each factor to zero.

The reason it works: if two things multiply to zero, at least one of them is zero. That is true of nothing except zero, which is why the equation has to be arranged with zero on a side first.

Solve x squared + 5x = 14

• Move everything to one side: x squared + 5x - 14 = 0.
• Factor: two numbers multiplying to -14 and adding to 5 are 7 and -2, so (x + 7)(x - 2) = 0.
• Set each to zero: x + 7 = 0 gives x = -7; x - 2 = 0 gives x = 2.

Solutions: x = -7 and x = 2.

A quadratic usually has two solutions and the test usually wants both. If only one appears among the choices, that one is the answer; if the question asks for "the positive solution", read it carefully.

Do not divide both sides by x to solve x squared = 5x. That loses the solution x = 0. Move everything to one side instead: x squared - 5x = 0 factors to x(x - 5) = 0, giving x = 0 and x = 5.

When it will not factor: the quadratic formula

For ax squared + bx + c = 0:

x = (-b plus or minus the square root of (b squared - 4ac)) / 2a

Solve 2x squared - 5x - 3 = 0.

a = 2, b = -5, c = -3. The part under the root, b squared - 4ac, is 25 + 24 = 49, and its root is 7. So x = (5 plus or minus 7) / 4, which gives 3 or -1/2.

The part under the root is the discriminant. Positive gives two solutions, zero gives one, negative gives none that are real numbers. For x squared - 4x + 5, it is 16 - 20 = -4.

Five questions say "using the quadratic formula", but every one of them also factors. Factoring is usually faster; the formula is the fallback.

What you can skip

Across the 77 questions on this topic:

• Completing the square as a method never appears.
• Sums and differences of cubes. The cubic questions pull out a common x first and then factor an ordinary quadratic.

Where people lose points

Forgetting the middle term when squaring a binomial.

Trying to factor a sum of squares.

Not pulling out the common factor first, which leaves numbers too big to factor by inspection.

Stopping before the factorization is complete.

Getting a sign wrong in the trinomial pair. The sign table above prevents most of it.

Giving one solution to a quadratic when there are two.

Dividing by the variable, which deletes a solution.

Work one in under a minute

Solve 2x squared - 8x - 24 = 0.

Pull out the common factor of 2 first: 2(x squared - 4x - 12) = 0.

Divide both sides by 2, which is safe because 2 is a constant: x squared - 4x - 12 = 0.

Factor: c is negative so the signs differ; two numbers multiplying to -12 and adding to -4 are -6 and 2.

(x - 6)(x + 2) = 0, so x = 6 and x = -2.

Check: 36 minus 24 minus 12 is 0. Correct.

Where this leads

Factoring is how you simplify algebraic fractions and how most quadratics on this test get solved without a formula.