PSAT Advanced Math — quadratics, and which form shows what

A quadratic can be written three ways, and the whole of this part of the test is knowing which one to reach for. Standard form shows the constant — the starting height of a thrown ball, the revenue at no price increase. Factored form shows the roots — the moment the ball lands. Vertex form shows the turning point — the greatest height, or the number of units that makes a profit largest. Choosing the form is usually the question; the algebra that follows is small.

This material is twelve questions arranged around that idea. Three are factoring, with whole-number roots throughout, including one with a leading coefficient that is not one and one difference of squares. Every wrong option in these was expanded by hand before it was kept, so that no two of them are the same expression in disguise and each is the result of a mistake that can be named: factors that add to the wrong number, a sign kept where it should have flipped, a square written where a product belongs.

Three ask directly which feature a given form displays, including one that asks what can be read off a vertex-form parabola with no calculation at all — where the wrong options are not false but merely one substitution away, which is the distinction the question is built on.

Two are vertex questions set in situations, and both turn on a trap worth meeting early: the vertex has two coordinates, and the wrong answer is nearly always the other one. A profit model peaks at five units and nine dollars; a fence encloses the most area at a width of ten meters and an area of one hundred. Read the question to the end before choosing.

Two put roots in a situation that rejects one of them — a projectile whose height is zero at seven seconds and at minus one, a revenue model that is zero both at no price increase and at six dollars more. The explanations say which root is thrown out and why, because a root that is arithmetically correct and physically impossible is the commonest way to lose one of these.

The last two are the discriminant: how many real solutions a quadratic has, and what value of a constant gives it exactly one. Neither requires quoting the quadratic formula.

There are no figures and no graphs here; every expression is written out in the question. All the situations were invented for this exercise, and nothing is reproduced from any published test.

• Choose the form of a quadratic that already contains what the question asks for
• Factor a quadratic with whole-number roots, including one with a leading coefficient other than one
• Recognize a difference of squares and factor it in one step
• Find a vertex from standard form, and say which of its two coordinates the question wants
• Reject the root a situation forbids, and say why it is impossible rather than merely unwanted
• Use the discriminant to count real solutions, and set it to zero to force exactly one
• Spot the wrong option that is one substitution away from being right, rather than false

Every expression and situation here was written for this exercise; there are no figures, and each question carries its own mathematics. Among them: factoring $x^2 + 7x + 10$ and $2x^2 + 5x + 3$; solving $x^2 - 9x + 18 = 0$; recognizing $x^2 - 16$ as a difference of squares; the thrown ball $h(t) = -t^2 + 4t + 5$, whose landing time only the factored form displays; the parabola $y = (x - 3)^2 + 4$, where the vertex is free and everything else costs a substitution; the profit model $P(x) = -x^2 + 10x - 16$, greatest at five units and nine dollars; the enclosed area $A(w) = -w^2 + 20w$; the revenue $R(x) = -2x^2 + 12x$, zero at no increase and at six dollars more; and $x^2 + 6x + k = 0$, which has one solution when $k$ is nine. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the PSAT/NMSQT, and it is not an exam centre.