Quadratic Equations — 700-800

Quadratic Equations: The 700-800 Patterns

Part 1 of 3 — The Archetypes Hard-Tier Items Are Built From

At the 700-800 level, quadratics questions stop asking you to solve equations and start asking you to reason about them. Three archetypes cover almost every hard item.

Archetype 1: Discriminant With a Parameter

A line and a parabola "intersect at exactly one point," or an equation "has no real solutions," and a constant ($k$, $m$, $b$, $c$) is unknown. The move is always the same:

1. Set the two expressions equal and move everything to one side.
2. Write the discriminant $b^{2} - 4ac$ in terms of the parameter.
3. Set it $= 0$ (tangent / one solution), $> 0$ (two), or $< 0$ (none).

Worked example. The line $y = 8x + b$ is tangent to $y = 2x^{2} - 12x + 19$. Setting equal: $2x^{2} - 20x + (19 - b) = 0$. Tangency means discriminant zero: $400 - 8(19 - b) = 0$, so $400 = 152 - 8b$ and $b = -31$.

The 700-800 twist: the question rarely asks for the parameter itself. It asks for the $x$- or $y$-coordinate of the tangent point, or a value built from it. Read the last sentence twice — every intermediate value ($m$, then $x$, then $y$) appears among the choices.

Archetype 2: Vertex Optimization With a Hidden Conversion

Max height, max revenue, min cost. Vertex at $t = -\frac{b}{2a}$, then substitute back. Hard-tier versions bury one extra step:

• Units: "$x$ hundred units" — the vertex $x = 7$ means the answer is $700$.
• Build the model yourself: "each $2 fee increase loses 10 members" means revenue is $(40 + 2n)(300 - 10n)$ — expand, then vertex.
• Whole-number restriction: if the vertex lands at $n = 3.75$ but $n$ must be a whole number, test $n = 3$ AND $n = 4$. The vertex value itself is unattainable and will be a choice.

Archetype 3: Vieta's Shortcuts (Sum and Product of Roots)

For $ax^{2} + bx + c = 0$ with roots $r$ and $s$:

• $r + s = -\frac{b}{a}$
• $rs = \frac{c}{a}$

Hard items ask for expressions built from the roots so that you never need the roots themselves:

• $\frac{1}{r} + \frac{1}{s} = \frac{r + s}{rs} = -\frac{b}{c}$
• $r^{2} + s^{2} = (r + s)^{2} - 2rs$
• $(r - s)^{2} = (r + s)^{2} - 4rs$

Worked example. For $3x^{2} + 7x - 6 = 0$: $\frac{1}{r} + \frac{1}{s} = \frac{-7/3}{-2} = \frac{7}{6}$. Ten seconds, no quadratic formula, no radicals.

Sign discipline is the whole game: two of the four answer choices will differ from the answer only by a dropped negative.

Quadratic Equations: Traps & Speed

Part 2 of 3 — Distractor Autopsy and Faster Routes

Every hard-tier quadratic item plants its wrong answers deliberately. Learn the four plants and you can often eliminate two choices before doing any algebra.

The Four Standard Distractors

1. The intermediate value. The time when the max occurs (asked: max height). The optimal fee (asked: max revenue). The value of $a$ (asked: $b$). Whatever value your work naturally produces FIRST is sitting in the choices.
2. The unit trap. "$x$ hundred units" — the raw vertex value $7$ appears next to $700$. "Thousands of dollars" appears next to dollars.
3. The sign flip. In Vieta problems, the answer with one negative dropped is always a choice.
4. The unattainable vertex. When the input must be a whole number and the vertex is not, the exact vertex value $R(3.75)$ is planted. Test both neighboring integers instead.

Speed Techniques

• Factored form is a gift: $P(x) = -2(x - 3)(x - 11)$ has its vertex midway between the zeros, at $x = 7$. Never expand.
• Vieta before formula: any question about $r + s$, $rs$, $\frac{1}{r} + \frac{1}{s}$, or $r^{2} + s^{2}$ is a 10-second Vieta computation. If your scratch work has a $\sqrt{\phantom{x}}$ in it, you took the slow road.
• Desmos exploit: for tangency problems, type the parabola and $y = mx + 4$ with a slider for $m$ — slide until the intersection points merge. For max-revenue models, type the revenue expression in one variable and click the vertex. Desmos gives exact coordinates.
• Answer-choice arithmetic: on "greatest revenue" items, the current (unoptimized) revenue is always a choice — compute it first and eliminate everything at or below it.

Quadratic Equations: Timed Drill

Part 3 of 3 — Four Questions at Full Difficulty

Set a pace of about 75 seconds per question — that is the real budget for a hard Module 2 item. For each question: identify the archetype in the first 10 seconds, pick the fast route (Vieta, symmetry, discriminant), and check that your final number answers the question actually asked before you click.

If you finish one early, spend the leftover seconds on the sanity check, not on the next question.