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Monday — Discriminants + Radical Simplification
Tuesday — Quadratic Formula End-to-End
Wednesday — Choose Your Method: Factoring vs. Formula
Thursday — Arithmetic & Geometric Sequences
Friday — Mixed Part II / III Short Response
Saturday — Full Part II + Part III Timed

Week 2 — Quadratics + Sequences End-to-End

Goal this week: finish what you start. Quadratic formula all the way to simplest radical form, and arithmetic sequences without off-by-one errors.

Time: ~60 min weekdays, ~75 min Saturday.

Heuristics in heavy use: H3 (count gaps not dots), H4 (simplest form), H5 (re-do heavy arithmetic).

How to use the practice problems: every problem has a Hint and a Show answer drop-down. Attempt the problem on paper first; open Hint only if you’re stuck; reveal Show answer only after you’ve committed to one of your own.

Monday — Discriminants + Radical Simplification

Objective

Build instant fluency with the discriminant $b^2 - 4ac$ and radical simplification. This is the most error-prone single line in the quadratic formula (Q32 on June 25).

Instructions

Memorize the radical-simplification table. Cover the right column and quiz yourself until you can answer all 12 in under 30 seconds.
Do all 8 discriminant problems. Compute $b^2 - 4ac$ , then simplify the radical, but stop there — don’t solve the full quadratic yet.
For every problem, recompute the discriminant on a side line (H5).

Radical-simplification table (memorize)

$n$	$\sqrt{n}$	$n$	$\sqrt{n}$
8	$2\sqrt{2}$	32	$4\sqrt{2}$
12	$2\sqrt{3}$	45	$3\sqrt{5}$
18	$3\sqrt{2}$	48	$4\sqrt{3}$
20	$2\sqrt{5}$	50	$5\sqrt{2}$
27	$3\sqrt{3}$	72	$6\sqrt{2}$
28	$2\sqrt{7}$	75	$5\sqrt{3}$

Pattern hint: find the largest perfect square that divides $n$ (4, 9, 16, 25, 36, 49, 64, 81, 100), take its root outside, leave the rest inside.

Practice problems — discriminant only

For each quadratic $ax^2 + bx + c = 0$ , identify $a, b, c$ , compute the discriminant, and simplify the radical.

1. $x^2 + 6x + 5 = 0$

Hint

$a = 1$ , $b = 6$ , $c = 5$ . Discriminant $= b^2 - 4ac$ .

Show answer

$a, b, c = 1, 6, 5$ . Disc $= 36 - 20 = 16$ . $\sqrt{16} = 4$ .

2. $2x^2 + 4x - 3 = 0$

Hint

Watch the negative $c$ : $-4ac = -4(2)(-3) = +24$ .

Show answer

$a, b, c = 2, 4, -3$ . Disc $= 16 + 24 = 40$ . $\sqrt{40} = 2\sqrt{10}$ .

3. $3x^2 - 6x - 1 = 0$

Hint

$b$ is negative; $b^2$ is positive. $-4(3)(-1) = +12$ .

Show answer

$a, b, c = 3, -6, -1$ . Disc $= 36 + 12 = 48$ . $\sqrt{48} = 4\sqrt{3}$ .

4. $x^2 + 2x - 7 = 0$

Hint

$-4ac = -4(1)(-7) = +28$ .

Show answer

$a, b, c = 1, 2, -7$ . Disc $= 4 + 28 = 32$ . $\sqrt{32} = 4\sqrt{2}$ .

5. $6x^2 + 2x - 1 = 0$ (the original Q32)

Hint

$-4ac = -4(6)(-1) = +24$ .

Show answer

$a, b, c = 6, 2, -1$ . Disc $= 4 + 24 = 28$ . $\sqrt{28} = 2\sqrt{7}$ .

6. $4x^2 - 4x - 1 = 0$

Hint

$b^2 = 16$ and $-4ac = +16$ .

Show answer

$a, b, c = 4, -4, -1$ . Disc $= 16 + 16 = 32$ . $\sqrt{32} = 4\sqrt{2}$ .

7. $5x^2 + 10x + 1 = 0$

Hint

$b^2 = 100$ and $-4ac = -20$ .

Show answer

$a, b, c = 5, 10, 1$ . Disc $= 100 - 20 = 80$ . $\sqrt{80} = 4\sqrt{5}$ .

8. $2x^2 - 3x - 4 = 0$

Hint

$9 - 4(2)(-4) = 9 + 32 = 41$ . Is $41$ prime?

Show answer

$a, b, c = 2, -3, -4$ . Disc $= 9 + 32 = 41$ . $\sqrt{41}$ — prime, leave as-is.

Self-check question

What does it mean when the discriminant is negative? (No real solutions; the parabola doesn’t cross the $x$ -axis.)

Tuesday — Quadratic Formula End-to-End

Objective

Carry the quadratic formula through all six steps to simplest radical form, every time. This directly fixes Q32.

The 6-step checklist (tape this above your desk)

Write $a$ , $b$ , $c$ on their own lines.
Substitute into $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with parentheses around negatives.
Compute discriminant on a separate line. Recompute (H5).
Simplify the radical (largest perfect-square factor).
Reduce the fraction (factor a GCF out of numerator AND denominator).
Write both answers (or use $\pm$ ).

Practice problems — full end-to-end

Solve each. Write your final answer in simplest radical form.

1. $x^2 + 6x + 4 = 0$

Hint

Disc $= 36 - 16 = 20 = 4 \cdot 5$ .

Show answer

$x = -3 \pm \sqrt{5}$ .

$x = \dfrac{-6 \pm \sqrt{20}}{2} = \dfrac{-6 \pm 2\sqrt{5}}{2} = -3 \pm \sqrt{5}$ .

2. $2x^2 + 4x - 3 = 0$

Hint

Disc $= 40$ ; the numerator/denominator won’t share a common factor here.

Show answer

$x = \dfrac{-2 \pm \sqrt{10}}{2}$ .

$x = \dfrac{-4 \pm \sqrt{40}}{4} = \dfrac{-4 \pm 2\sqrt{10}}{4} = \dfrac{-2 \pm \sqrt{10}}{2}$ .

3. $3x^2 - 6x - 1 = 0$

Hint

Disc $= 48 = 16 \cdot 3$ ; numerator and denominator share a factor of 2.

Show answer

$x = \dfrac{3 \pm 2\sqrt{3}}{3}$ .

$a = 3, b = -6, c = -1$ . $x = \dfrac{-(-6) \pm \sqrt{36 + 12}}{6} = \dfrac{6 \pm \sqrt{48}}{6} = \dfrac{6 \pm 4\sqrt{3}}{6} = \dfrac{3 \pm 2\sqrt{3}}{3}$ .

4. $6x^2 + 2x - 1 = 0$ (the original Q32)

Hint

Disc $= 28 = 4 \cdot 7$ ; factor 2 out of numerator and denominator at the end.

Show answer

$x = \dfrac{-1 \pm \sqrt{7}}{6}$ .

$a = 6, b = 2, c = -1$ . Disc $= 4 + 24 = 28$ . $\sqrt{28} = 2\sqrt{7}$ . $x = \dfrac{-2 \pm 2\sqrt{7}}{12} = \dfrac{-1 \pm \sqrt{7}}{6}$ .

5. $4x^2 - 4x - 1 = 0$

Hint

Disc $= 32 = 16 \cdot 2$ ; reduce by a factor of 2.

Show answer

$x = \dfrac{1 \pm \sqrt{2}}{2}$ .

$x = \dfrac{4 \pm \sqrt{32}}{8} = \dfrac{4 \pm 4\sqrt{2}}{8} = \dfrac{1 \pm \sqrt{2}}{2}$ .

6. $5x^2 + 10x + 1 = 0$

Hint

Disc $= 80 = 16 \cdot 5$ ; reduce by 2.

Show answer

$x = \dfrac{-5 \pm 2\sqrt{5}}{5}$ .

$x = \dfrac{-10 \pm \sqrt{80}}{10} = \dfrac{-10 \pm 4\sqrt{5}}{10} = \dfrac{-5 \pm 2\sqrt{5}}{5}$ .

Self-check question

After step 4, why might you still need step 5? (Because the GCF in the numerator may share a factor with the denominator — common when $b$ is even and $a$ has a factor of 2.)

Wednesday — Choose Your Method: Factoring vs. Formula

Objective

Pick the cheapest tool for the job. Not every quadratic deserves the full formula.

Decision rule

Try factoring first if $a = 1$ and you can find two integers that multiply to $c$ and add to $b$ in under 15 seconds.
Use the formula if factoring doesn’t pop in 15 seconds, or if $a \neq 1$ and the leading coefficient is awkward.
Complete the square is rarely tested for Regents end-to-end solving — skip unless explicitly asked.

Practice problems

For each, decide F (factor) or Q (quadratic formula), justify in one sentence, then solve.

1. $x^2 - 5x + 6 = 0$

Hint

$a = 1$ ; find two integers that multiply to $6$ and add to $-5$ .

Show answer

Method: F. $(x - 2)(x - 3) = 0$ . $x = 2, 3$ .

2. $x^2 + 7x + 10 = 0$

Hint

$2 \cdot 5 = 10$ and $2 + 5 = 7$ — but check signs.

Show answer

Method: F. $(x + 2)(x + 5) = 0$ . $x = -2, -5$ .

3. $2x^2 + 5x - 3 = 0$

Hint

AC method: $AC = -6$ . Find two numbers multiplying to $-6$ and adding to $+5$ : $+6$ and $-1$ .

Show answer

Method: F (AC) or Q. $x = \tfrac{1}{2}, -3$ .

Split the middle: $2x^2 + 6x - x - 3 = 2x(x + 3) - 1(x + 3) = (2x - 1)(x + 3)$ .

4. $x^2 - 4x - 7 = 0$

Hint

Doesn’t factor over the integers (no two integers multiply to $-7$ and add to $-4$ ). Use the formula.

Show answer

Method: Q. Disc $= 16 + 28 = 44 = 4 \cdot 11$ . $x = \dfrac{4 \pm 2\sqrt{11}}{2} = 2 \pm \sqrt{11}$ .

5. $x^2 + 8x + 16 = 0$

Hint

PST: $16 = 4^2$ and $8 = 2(4)$ .

Show answer

Method: F (PST). $(x + 4)^2 = 0$ . $x = -4$ (double root).

6. $3x^2 + 4x - 2 = 0$

Hint

Doesn’t factor nicely; use the formula. Disc $= 16 + 24 = 40$ .

Show answer

Method: Q. $x = \dfrac{-4 \pm \sqrt{40}}{6} = \dfrac{-4 \pm 2\sqrt{10}}{6} = \dfrac{-2 \pm \sqrt{10}}{3}$ .

Self-check question

How can you check a factored quadratic? (Multiply the factors back out — should equal the original. Or, plug each root into the original equation.)

Thursday — Arithmetic & Geometric Sequences

Objective

Eliminate the off-by-one error that cost a point on Q28 (used $d = 9$ instead of $d = 6$ ). Always draw the dot-and-gap diagram first (H3).

The two formulas

Arithmetic: $a_n = a_1 + (n - 1)\,d$ where $d = a_{n+1} - a_n$ (common difference).

Geometric: $a_n = a_1 \cdot r^{\,n-1}$ where $r = a_{n+1}/a_n$ (common ratio).

The dot-and-gap diagram

For “find $d$ when $a_1$ and $a_k$ are given”:

a_1 \xrightarrow{d} a_2 \xrightarrow{d} \cdots \xrightarrow{d} a_k

Count the arrows (= $k - 1$ ). Then $(k - 1)\,d = a_k - a_1$ .

Practice problems

1. Arithmetic: $a_1 = -20$ , $a_4 = -2$ . Find $a_8$ . (Q28)

Hint

3 gaps from $a_1$ to $a_4$ , so $3d = -2 - (-20) = 18$ . Then $a_8 = a_1 + 7d$ .

Show answer

$d = 6$ , $a_8 = -20 + 7(6) = 22$ .

Mistake on June 25: counted dots (4) instead of gaps (3), got $d = 9$ instead of $d = 6$ .

2. Arithmetic: $a_1 = 7$ , $a_5 = 23$ . Find $a_{10}$ .

Hint

4 gaps from $a_1$ to $a_5$ : $4d = 16$ .

Show answer

$d = 4$ , $a_{10} = 7 + 9(4) = 43$ .

3. Arithmetic: $a_3 = 11$ , $a_8 = 36$ . Find $a_1$ .

Hint

5 gaps between $a_3$ and $a_8$ , so $5d = 25$ . Then walk back: $a_1 = a_3 - 2d$ .

Show answer

$d = 5$ , $a_1 = 11 - 10 = 1$ .

4. Arithmetic: $a_1 = 100$ , $d = -7$ . Find $a_{15}$ .

Hint

$a_{15} = a_1 + 14d$ .

Show answer

$a_{15} = 100 + 14(-7) = 2$ .

5. Geometric: $a_1 = 3$ , $r = 2$ . Find $a_6$ .

Hint

$a_6 = a_1 \cdot r^5$ .

Show answer

$a_6 = 3 \cdot 32 = 96$ .

6. Geometric: 12, 6, 3, $\tfrac{3}{2}$ , … find $a_n$ formula.

Hint

Each term is half the previous, so $r = \tfrac{1}{2}$ .

Show answer

$a_n = 12 \cdot \left(\tfrac{1}{2}\right)^{n-1}$ .

7. Geometric: $a_1 = 5$ , $a_4 = 40$ . Find $r$ and $a_6$ .

Hint

$a_4 / a_1 = r^3 = 8$ .

Show answer

$r = 2$ , $a_6 = 5 \cdot 32 = 160$ .

8. Arithmetic: $a_1 = -3$ , $a_{10} = 24$ . Find $d$ .

Hint

9 gaps between $a_1$ and $a_{10}$ , so $9d = 27$ .

Show answer

$d = 3$ .

9. Geometric: $a_2 = 18$ , $a_5 = 486$ . Find $a_1$ and $r$ .

Hint

$a_5 / a_2 = r^3$ (3 ratio gaps between $a_2$ and $a_5$ ).

Show answer

$r^3 = 27 \implies r = 3$ . $a_1 = a_2 / r = 18 / 3 = 6$ .

10. Arithmetic: 8th term is 30, 12th term is 50. Find $a_1$ .

Hint

4 gaps between $a_8$ and $a_{12}$ , so $4d = 20$ . Then $a_1 = a_8 - 7d$ .

Show answer

$d = 5$ , $a_1 = 30 - 35 = -5$ .

Self-check question

If you’re told $a_3 = 11$ and $a_8 = 36$ , how many gaps are between them? (5 gaps, not 6 — that’s the count $8 - 3$ .)

Friday — Mixed Part II / III Short Response

Objective

Apply the heuristics under timed short-response conditions. Self-grade strictly.

Instructions

Set timer for 30 minutes.
Solve all 6 problems with full work shown — no hints or answers during the timed set.
Use the 6-step quadratic checklist and the dot-and-gap diagram where applicable.
After the timer ends, self-grade using the rubric below; only then reveal answers.

Practice problems

1. (2 pts) Express $(3x - 4)(2x + 5)$ as a trinomial in standard form.

Hint

Box method, watch signs.

Show answer

$6x^2 + 7x - 20$ .

2. (2 pts) Solve $4(x - 2) > -2x + 14$ for $x$ .

Hint

Distribute first: $4x - 8 > -2x + 14$ . Collect $x$ ‘s on one side; no sign-flip if you keep $x$ positive.

Show answer

$x > \tfrac{11}{3}$ . ( $6x > 22 \implies x > \tfrac{11}{3}$ .)

3. (2 pts) The first and third terms of an arithmetic sequence are 6 and 20. Determine the seventh term.

Hint

2 gaps between $a_1$ and $a_3$ : $2d = 14$ .

Show answer

$d = 7$ , $a_7 = 6 + 6(7) = 48$ .

4. (4 pts) Using the quadratic formula, solve $3x^2 - 4x - 2 = 0$ . Express the answer in simplest radical form.

Hint

Disc $= 16 + 24 = 40 = 4 \cdot 10$ . Reduce by a factor of 2 at the end.

Show answer

$x = \dfrac{2 \pm \sqrt{10}}{3}$ .

$a = 3, b = -4, c = -2$ . $x = \dfrac{4 \pm \sqrt{40}}{6} = \dfrac{4 \pm 2\sqrt{10}}{6} = \dfrac{2 \pm \sqrt{10}}{3}$ .

5. (4 pts) Factor completely: $4x^3 - 100x$ .

Hint

GCF $4x$ , then DOS on $x^2 - 25$ .

Show answer

$4x(x - 5)(x + 5)$ .

6. (4 pts) The table below shows hours studied $x$ and quiz score $y$ .

$x$	1	2	3	4	5
$y$	60	68	73	82	90

State the linear regression equation rounded to the nearest hundredth, the correlation coefficient rounded to the nearest hundredth, and what the correlation coefficient indicates.

Hint

On your calculator: STAT → CALC → LinReg(ax+b). Make sure DiagnosticOn is set to see $r$ .

Show answer

$y \approx 7.40x + 53.40$ ; $r \approx 0.99$ ; strong positive correlation between hours studied and quiz score.

Self-grade rubric

For each problem, award:

Full credit if final answer matches AND all work is shown.
1 less if final answer right but a step is missing.
1 less if computational error but method is right.
0 if method is fundamentally wrong.

Target: 14+ / 18.

Saturday — Full Part II + Part III Timed

Objective

Measure progress on the highest point-value section (Part III was the weakest area on June 25).

Instructions

Pull Part II (Q25–Q30) and Part III (Q31–Q34) from a prior Regents.
Timer: 75 minutes (Part II ≈ 30 min, Part III ≈ 45 min).
Work in pen. Use scratch paper for calculations, neat work on the test.
Apply the 6-step quadratic checklist and the dot-and-gap diagram everywhere relevant.
Score against the official Model Response Set.

Success bar for Week 2

Part II score: 11–12/12.
Part III score: 10+/16 (vs. ~5/16 baseline).
Zero quadratic-formula answers left in unsimplified form.
Zero off-by-one errors on sequence problems.

← Back to main study plan | ← Week 1 | → Week 3