How do you factor a quadratic expression on the GRE?

First pull out any common factor. Then, for x² + bx + c, find two numbers that multiply to c and add to b; those numbers become the constants in (x + p)(x + q). Watch for a difference of squares, a² - b² = (a + b)(a - b), which factors instantly.

How do you set up a mixture problem?

Track the amount of the pure substance, which equals concentration times volume. The pure amounts are conserved when solutions combine, so set the sum of the parts equal to the concentration of the final mixture times its total volume, then solve.

Why can’t you just average the two times in a work problem?

Because work adds as rates, not as times. Two workers who each take 10 and 15 hours have rates 1/10 and 1/15; adding those gives 1/6, so together they finish in 6 hours — faster than either alone, which averaging the times would never show.

What is the order trap in translating "less than"?

The phrase "A less than B" translates to B - A, not A - B. So "7 less than x" is x - 7. Reading the subtraction in writing order is one of the most common GRE translation mistakes.

CramClub • Algebraic Expressions and Word Problems — GRE® General Test Guide & Practice

$(a + b)(a - b) = a^2 - b^2$

Two numbers with product $c$ and sum $b$

$I = P \cdot r \cdot t$

$\dfrac{1}{t_{\text{together}}} = \dfrac{1}{t_A} + \dfrac{1}{t_B}$

From Words to Algebra

The hardest part of a GRE algebra word problem is almost never the algebra — it is the translation. Once a mixture, interest, age, or work scenario is written as an equation, the solving is routine. This guide builds two linked skills: manipulating expressions (simplifying, expanding with FOIL, and factoring) and modeling (turning English into equations for the GRE's four classic word-problem families). Both reward precision over speed.

Simplifying and Combining Like Terms

An expression has no equals sign; you simplify it, you do not "solve" it. Combine only like terms — terms with the same variable raised to the same power.

$3x^2 + 5x - x^2 + 2 - 4x = 2x^2 + x + 2.$

Distribute a negative across a full group carefully: $7 - (2x - 3) = 7 - 2x + 3 = 10 - 2x$ . The sign error of writing $7 - 2x - 3$ is one of the most common on the whole test.

Expanding with FOIL

To multiply two binomials, multiply every term in the first by every term in the second — First, Outer, Inner, Last:

$(2x - 3)(x + 5) = 2x^2 + 10x - 3x - 15 = 2x^2 + 7x - 15.$

Memorize the three special products; the GRE plants them constantly:

Pattern	Expansion
$(a + b)^2$	$a^2 + 2ab + b^2$
$(a - b)^2$	$a^2 - 2ab + b^2$
$(a + b)(a - b)$	$a^2 - b^2$

The difference of squares is a genuine shortcut: $(x + 7)(x - 7) = x^2 - 49$ needs no middle term.

Factoring — Running the Machine Backward

Factoring reverses expansion. First pull out any common factor: $6x^2 + 9x = 3x(2x + 3)$ . To factor $x^2 + bx + c$ , find two numbers that multiply to $c$ and add to $b$ . For $x^2 - 5x + 6$ , the numbers $-2$ and $-3$ work, so $x^2 - 5x + 6 = (x - 2)(x - 3)$ . Spotting a difference of squares — $x^2 - 49 = (x - 7)(x + 7)$ — turns a nasty-looking expression into two clean factors.

Translating the Four Word-Problem Families

1. Age. Track each person's age now and shift by the same amount for past or future. If a father is 3 times his son's age now and will be twice as old in 12 years, then with son $= s$ : $3s + 12 = 2(s + 12)$ .

2. Mixture. The amount of the pure substance is concentration $\times$ volume, and it is conserved when you combine. Adding $x$ liters of a 20% solution to 10 liters of a 50% solution to reach 30% gives $0.20x + 0.50(10) = 0.30(x + 10)$ .

3. Simple interest. Interest $= P \cdot r \cdot t$ . Splitting a principal across two rates, the interests add: if $8000 is split so that $x$ earns 8% and the rest earns 5% for one year, total interest is $0.08x + 0.05(8000 - x)$ .

4. Work / rate. Work done $=$ rate $\times$ time, and rates add. If pipe A fills a tank in 6 hours (rate $\tfrac{1}{6}$ ) and pipe B in 4 hours (rate $\tfrac{1}{4}$ ), together their rate is $\tfrac{1}{6} + \tfrac{1}{4} = \tfrac{5}{12}$ , so the time is the reciprocal, $\tfrac{12}{5}$ hours.

A Translation Dictionary

English	Algebra
"is", "was", "will be"	$=$
"more than", "increased by"	$+$
"less than", "fewer than"	$-$ (mind the order!)
"of", "product", "times"	$\times$
"per", "for each"	$\div$ or a rate
"consecutive even integers"	$n,\; n + 2,\; n + 4$

The order trap in the second row is worth memorizing: "5 less than $x$ " is $x - 5$ , not $5 - x$ .

A Reliable Game Plan

Simplify? Combine like terms only; distribute negatives across the whole group.
Expand? FOIL every pair; recognize $(a \pm b)^2$ and $(a+b)(a-b)$ on sight.
Factor? Pull out common factors first, then find two numbers multiplying to $c$ and adding to $b$ .
Word problem? Name a variable, write each fact as an equation using the dictionary above, and check that units match before you solve.

Simplify $5x^2 - 2x + 3 - (2x^2 - 5x + 8)$ .

Expand $(3x - 2)(2x + 5)$ .

Factor $x^2 - 3x - 28$ completely.

Maria is 4 times as old as her nephew. In 5 years, she will be 3 times as old as he will be. How old is Maria now?

An investor splits $8,000 between an account paying 5% and one paying 8% simple annual interest. After one year the total interest is $520. How much was invested at 8%?

One printer can finish a job in 10 hours and a second printer can finish it in 15 hours. Working together, how long do they take to finish the job?

Distributing a subtraction to only the first term: $7 - (2x - 3) = 7 - 2x - 3$ .

A minus sign flips every term in the group: $7 - (2x - 3) = 7 - 2x + 3 = 10 - 2x$ .

Reversing the order for "less than": writing "7 less than $x$ " as $7 - x$ .

" $A$ less than $B$ " means $B - A$ . So "7 less than $x$ " is $x - 7$ . Read the phrase right to left.

Averaging or adding times in a work problem — treating two printers of 10 and 15 hours as 12.5 hours together.

Add the rates, not the times: $\tfrac{1}{10} + \tfrac{1}{15} = \tfrac{1}{6}$ , so together it is 6 hours, faster than either alone.

Dropping a middle term when squaring a binomial: writing $(x + 4)^2 = x^2 + 16$ .

Use $(a + b)^2 = a^2 + 2ab + b^2$ . Here $(x + 4)^2 = x^2 + 8x + 16$ ; the $2ab$ term is not optional.

Which of the following is equivalent to $(2x - 3)(x + 5)$ ?

$x$ is a nonzero number.

Quantity A: $x^2$

Quantity B: $x^3$

A father is currently 3 times as old as his son. In 12 years, he will be only twice as old as his son will be. How old is the son now?

How many liters of a 20% acid solution must be added to 10 liters of a 50% acid solution to produce a 30% acid solution?

Pipe A can fill a tank in 6 hours and pipe B can fill the same tank in 4 hours. If both pipes run together, how long will it take to fill the tank?

The sum of three consecutive even integers is 78. What is the greatest of the three integers?