Week 2 — Quadratics + Sequences End-to-End

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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).

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.

#	Equation	$a, b, c$	Discriminant	$\sqrt{\text{disc}}$ simplified
1	$x^2 + 6x + 5 = 0$	$1, 6, 5$	$36 - 20 = 16$	$4$
2	$2x^2 + 4x - 3 = 0$	$2, 4, -3$	$16 + 24 = 40$	$2\sqrt{10}$
3	$3x^2 - 6x - 1 = 0$	$3, -6, -1$	$36 + 12 = 48$	$4\sqrt{3}$
4	$x^2 + 2x - 7 = 0$	$1, 2, -7$	$4 + 28 = 32$	$4\sqrt{2}$
5	$6x^2 + 2x - 1 = 0$ (Q32!)	$6, 2, -1$	$4 + 24 = 28$	$2\sqrt{7}$
6	$4x^2 - 4x - 1 = 0$	$4, -4, -1$	$16 + 16 = 32$	$4\sqrt{2}$
7	$5x^2 + 10x + 1 = 0$	$5, 10, 1$	$100 - 20 = 80$	$4\sqrt{5}$
8	$2x^2 - 3x - 4 = 0$	$2, -3, -4$	$9 + 32 = 41$	$\sqrt{41}$ (prime — leave it)

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.

#	Equation	Answer
1	$x^2 + 6x + 4 = 0$	$x = -3 \pm \sqrt{5}$
2	$2x^2 + 4x - 3 = 0$	$x = \dfrac{-2 \pm \sqrt{10}}{2}$
3	$3x^2 - 6x - 1 = 0$	$x = \dfrac{3 \pm 2\sqrt{3}}{3}$
4	$6x^2 + 2x - 1 = 0$ (Q32)	$x = \dfrac{-1 \pm \sqrt{7}}{6}$
5	$4x^2 - 4x - 1 = 0$	$x = \dfrac{1 \pm \sqrt{2}}{2}$
6	$5x^2 + 10x + 1 = 0$	$x = \dfrac{-5 \pm 2\sqrt{5}}{5}$

Worked solution for #4 (the original Q32)

$a = 6,\ b = 2,\ c = -1$ .

Substitute: $x = \dfrac{-(2) \pm \sqrt{(2)^2 - 4(6)(-1)}}{2(6)}$

Discriminant: $4 - 4(6)(-1) = 4 + 24 = 28$ . Recompute: $4 + 24 = 28$ ✓.

$x = \dfrac{-2 \pm \sqrt{28}}{12}$

Simplify radical: $\sqrt{28} = 2\sqrt{7}$ .

$x = \dfrac{-2 \pm 2\sqrt{7}}{12}$

Reduce: factor 2 out of numerator and denominator.

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

Worked solution for #3 (negative b)

$a = 3,\ b = -6,\ c = -1$ .

Substitute: $x = \dfrac{-(-6) \pm \sqrt{(-6)^2 - 4(3)(-1)}}{2(3)} = \dfrac{6 \pm \sqrt{36 + 12}}{6} = \dfrac{6 \pm \sqrt{48}}{6}$ .

$\sqrt{48} = 4\sqrt{3}$ .

$x = \dfrac{6 \pm 4\sqrt{3}}{6}$

Reduce: divide numerator and denominator by 2.

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

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, write F (factor) or Q (quadratic formula) in the margin, justify in one sentence, then solve.

#	Equation	Method	Solutions
1	$x^2 - 5x + 6 = 0$	F	$x = 2, 3$
2	$x^2 + 7x + 10 = 0$	F	$x = -2, -5$
3	$2x^2 + 5x - 3 = 0$	F (AC method) or Q	$x = -3, \tfrac{1}{2}$
4	$x^2 - 4x - 7 = 0$	Q (doesn’t factor over integers)	$x = 2 \pm \sqrt{11}$
5	$x^2 + 8x + 16 = 0$	F (PST)	$x = -4$ (double root)
6	$3x^2 + 4x - 2 = 0$	Q	$x = \dfrac{-2 \pm \sqrt{10}}{3}$

Worked solution for #3 (AC method)

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

AC = $2 \cdot (-3) = -6$ . Find two numbers that multiply to $-6$ and add to $+5$ : those are $+6$ and $-1$ .

Split the middle: $2x^2 + 6x - x - 3 = 0$ .

Group: $2x(x + 3) - 1(x + 3) = 0 \implies (2x - 1)(x + 3) = 0$ .

$x = \tfrac{1}{2}$ or $x = -3$ .

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 $d = a_{n+1} - a_n$ (the common difference).

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

#	Problem	Answer
1	Arithmetic: $a_1 = -20$ , $a_4 = -2$ . Find $a_8$ . (Q28)	$d = 6$ , $a_8 = 22$
2	Arithmetic: $a_1 = 7$ , $a_5 = 23$ . Find $a_{10}$ .	$d = 4$ , $a_{10} = 43$
3	Arithmetic: $a_3 = 11$ , $a_8 = 36$ . Find $a_1$ .	$d = 5$ , $a_1 = 1$
4	Arithmetic: $a_1 = 100$ , $d = -7$ . Find $a_{15}$ .	$a_{15} = 2$
5	Geometric: $a_1 = 3$ , $r = 2$ . Find $a_6$ .	$a_6 = 96$
6	Geometric: 12, 6, 3, $\tfrac{3}{2}$ , … find $a_n$ formula.	$a_n = 12 \cdot (\tfrac{1}{2})^{n-1}$
7	Geometric: $a_1 = 5$ , $a_4 = 40$ . Find $r$ and $a_6$ .	$r = 2$ , $a_6 = 160$
8	Arithmetic: $a_1 = -3$ , $a_{10} = 24$ . Find $d$ .	$d = 3$
9	Geometric: $a_2 = 18$ , $a_5 = 486$ . Find $a_1$ and $r$ .	$r = 3$ , $a_1 = 6$
10	Arithmetic: 8th term is 30, 12th term is 50. Find $a_1$ .	$d = 5$ , $a_1 = -5$

Worked solution for #3 (a₃ and a₈ given)

Diagram: $a_3 \xrightarrow{d} a_4 \xrightarrow{d} a_5 \xrightarrow{d} a_6 \xrightarrow{d} a_7 \xrightarrow{d} a_8$

5 arrows from $a_3$ to $a_8$ . So $5d = 36 - 11 = 25 \implies d = 5$ .

Then to get $a_1$ : $a_1 = a_3 - 2d = 11 - 10 = 1$ .

Worked solution for #9 (geometric, given non-first terms)

$a_5 / a_2 = r^3$ (3 ratio gaps from $a_2$ to $a_5$ ).

$486 / 18 = 27 \implies r^3 = 27 \implies r = 3$ .

$a_1 = a_2 / r = 18 / 3 = 6$ .

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.
Use the 6-step quadratic checklist and the dot-and-gap diagram where applicable.
Self-grade using the rubric below.

Practice problems

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

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

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

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

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

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.

Answer key

1. $6x^2 + 7x - 20$ .

2. $x > \tfrac{11}{3}$ (or equivalent). $4x - 8 > -2x + 14 \implies 6x > 22 \implies x > \tfrac{11}{3}$ .

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

4. $x = \dfrac{2 \pm \sqrt{10}}{3}$ . $a = 3, b = -4, c = -2$ . Disc $= 16 + 24 = 40$ . $\sqrt{40} = 2\sqrt{10}$ . $x = \dfrac{4 \pm 2\sqrt{10}}{6} = \dfrac{2 \pm \sqrt{10}}{3}$ .

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

6. $y \approx 7.40x + 53.40$ ; $r \approx 0.99$ ; 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.

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