Algebra
Core skills
Algebra questions ask you to manipulate expressions and solve equations and inequalities. Expect linear equations and inequalities in one and two variables, slope and intercepts, systems of two equations (substitution, elimination, and the conditions for zero or infinitely many solutions), polynomial arithmetic and factoring, quadratic equations solved by factoring or the quadratic formula, the discriminant, rational expressions, and translation of word problems into equations.
Traps to expect
- Inequalities flip. Multiplying or dividing both sides by a negative number reverses the inequality sign.
- Distributing a minus sign: $-(x - 4) = -x + 4$, not $-x - 4$.
- Systems: identical slopes with different intercepts mean no solution; the same line written twice means infinitely many.
- Discriminant $b^2 - 4ac$: positive gives two real roots, zero gives exactly one, negative gives none. Read "exactly one" as "discriminant equals zero."
- Dividing by a variable can discard valid solutions; factor instead of dividing.
- Quadratic roots satisfy sum $= -b/a$ and product $= c/a$ - often faster than solving.
Worked example
For what values of $k$ does $2x^2 + kx + 8 = 0$ have exactly one real solution? Set the discriminant to zero: $k^2 - 4(2)(8) = 0$, so $k^2 = 64$ and $k = 8$ or $k = -8$. Both must be reported - a classic trap is giving only the positive value.
Rule of thumb
Before solving, rewrite the equation with zero on one side. It makes factoring, the quadratic formula, and the discriminant all immediately available.