Math Practice — Algebra 2 & Geometry

Algebra 2 — 10 Core Problems

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Fill in your work on each line below the problem. Choose the answer — get it right for 🎉!

COMPLEX CONJUGATE ROOTS — "if a+bi is a root, so is a−bi"

A·01 HARD ★★★★☆

📖 Concept Reminder A polynomial with real coefficients always has complex roots in conjugate pairs. If \(3 - 2i\) is a root, then \(3 + 2i\) must also be a root.

A polynomial \(p(x)\) with real coefficients has roots \(3 - 2i\) and \(-1\).
What is the minimum degree of \(p(x)\), and which of the following must also be a root?

RATIONAL EXPONENTS — "\(a^{m/n} = (\sqrt[n]{a})^m\)"

A·02 TRICKY ★★★☆☆

📖 Worked Example \(8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4\). Watch out: negative bases with even denominators are undefined in ℝ.

Simplify completely. Which expression is NOT equivalent to \((-27)^{4/3}\)?

LOG LAWS — "log(AB)=logA+logB · log(A/B)=logA−logB · log(Aⁿ)=n·logA"

A·03 HARD ★★★★★

📖 Trap Alert ⚠️ \(\log(A+B) \neq \log A + \log B\). This is the most common mistake! Only products/quotients split.

If \(\log_3 x = 4\) and \(\log_3 y = -2\), find: \(\log_3\!\left(\dfrac{x^2 \sqrt{y}}{27}\right)\)

COMPLETING THE SQUARE — "add (b/2)² to both sides · vertex form: a(x−h)²+k"

A·04 TRICKY ★★★★☆

The quadratic \(f(x) = 2x^2 - 12x + 11\) is rewritten in vertex form \(a(x - h)^2 + k\).
What is the minimum value of \(f(x)\)?

INVERSE FUNCTIONS — "swap x and y, then solve for y · domain/range swap too"

A·05 HARD ★★★★☆

📖 Key Fact If \(f(a) = b\), then \(f^{-1}(b) = a\). The graphs of \(f\) and \(f^{-1}\) are reflections over \(y = x\).

Let \(f(x) = \dfrac{2x + 3}{x - 1}\), \(x \neq 1\).
Find \(f^{-1}(4)\).

ARITHMETIC SERIES — "Sₙ = n/2·(a₁+aₙ) · or Sₙ = n/2·(2a₁+(n−1)d)"

A·06 TRICKY ★★★★☆

The sum of the first \(n\) terms of an arithmetic sequence is \(S_n = 3n^2 + 5n\).
What is the 12th term \(a_{12}\)?

SYNTHETIC DIVISION — "remainder = f(c) by Remainder Theorem · factor if remainder=0"

A·07 HARD ★★★★★

Using the Remainder Theorem, when \(p(x) = x^4 - 3x^3 + ax^2 - 5x + 6\) is divided by \((x - 2)\), the remainder is \(-4\).
Find \(a\). Then determine whether \((x + 1)\) is a factor of \(p(x)\) with this value of \(a\).

GEOMETRIC SERIES (INFINITE) — "S = a₁/(1−r) valid only when |r| < 1"

A·08 TRICKY ★★★★☆

📖 Trap Alert ⚠️ Students often apply the formula when \(|r| \geq 1\) — the series diverges and has no finite sum!

An infinite geometric series has first term \(a_1 = 12\) and common ratio \(r = \dfrac{2}{3}\).
A second series has \(a_1 = k\) and \(r = -\dfrac{1}{4}\), with the same infinite sum.
Find \(k\).

CONIC SECTIONS — "circle: r²=x²+y² · ellipse: x²/a²+y²/b²=1 · hyperbola: x²/a²−y²/b²=1"

A·09 HARD ★★★★★

The equation \(9x^2 - 4y^2 - 36x + 8y - 4 = 0\) represents a conic.
After completing the square, identify the conic and find the center.

BINOMIAL THEOREM — "C(n,k)·aⁿ⁻ᵏ·bᵏ · the (k+1)th term has bᵏ"

A·10 HARD ★★★★★

📖 Formula \(\displaystyle(a+b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k\)
The term containing \(b^k\) is the \((k+1)\)th term.

Find the coefficient of \(x^3\) in the expansion of \(\left(2x - \dfrac{1}{x}\right)^7\).

Geometry — 10 Core Problems

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Draw a diagram in the margin! Geometry is always easier with a picture.

PARALLEL LINES — "alternate interior EQUAL · co-interior (same-side) SUPPLEMENTARY=180°"

G·01 TRICKY ★★★☆☆

Two parallel lines are cut by a transversal. One co-interior (same-side interior) angle measures \((3x + 15)°\) and the other measures \((5x - 7)°\).
Find \(x\) and both angle measures.

TRIANGLE SIMILARITY — "AA · SSS~ · SAS~ · CASTRATE: corresponding sides proportional"

G·02 HARD ★★★★☆

📖 Trap Alert ⚠️ When two triangles share an angle and have one pair of proportional sides, that is not automatically SAS~. You must verify the sides are adjacent to the same angle.

In \(\triangle ABC\), \(DE \parallel BC\) with \(D\) on \(\overline{AB}\) and \(E\) on \(\overline{AC}\).
If \(AD = 6\), \(DB = 4\), and \(BC = 15\), find \(DE\).

CIRCLE ARCS & ANGLES — "inscribed angle = ½ central angle · tangent-chord = ½ arc"

G·03 HARD ★★★★★

📖 Angle-Arc Rules • Angle inside circle (chords cross): \(\theta = \frac{1}{2}(\text{arc}_1 + \text{arc}_2)\)
• Angle outside circle (secants/tangents): \(\theta = \frac{1}{2}|\text{far arc} - \text{near arc}|\)

Two chords \(\overline{AB}\) and \(\overline{CD}\) intersect at point \(P\) inside a circle.
Arc \(AC = 80°\) and arc \(BD = 60°\).
Find the measure of \(\angle APD\).

PYTHAGOREAN THEOREM — "a²+b²=c² · special triangles: 30-60-90 & 45-45-90"

G·04 TRICKY ★★★★☆

In a right triangle, the altitude from the right angle to the hypotenuse has length \(h = 6\).
The two segments of the hypotenuse created by this altitude have lengths in the ratio \(1:4\).
Find the hypotenuse.

SURFACE AREA vs VOLUME — "SA scales with r² · V scales with r³ · double r → SA×4, V×8"

G·05 HARD ★★★★★

A cone and a cylinder share the same base radius \(r\) and the same height \(h\).
The slant height of the cone is \(l = \sqrt{r^2 + h^2}\).
If \(r = 3\) and \(h = 4\), find the ratio of the cone's total surface area to the cylinder's total surface area.

COORDINATE GEOMETRY — "midpoint=avg · distance=√(Δx²+Δy²) · slope=Δy/Δx"

G·06 TRICKY ★★★★☆

The vertices of a triangle are \(A(1, 2)\), \(B(5, -2)\), and \(C(-1, -4)\).
Is this triangle a right triangle? If so, identify the right angle vertex.

CIRCLE EQUATION — "(x−h)²+(y−k)²=r² · expand carefully to find center & radius"

G·07 HARD ★★★★☆

The equation \(x^2 + y^2 - 6x + 10y + 18 = 0\) represents a circle.
Find the center and radius, then determine how many \(x\)-intercepts the circle has.

TRIANGLE CONGRUENCE — "SSS·SAS·ASA·AAS = CONGRUENT · SSA/AAA = NOT enough (TRAP!)"

G·08 TRICKY ★★★★☆

📖 The SSA Trap SSA (two sides and a non-included angle) can produce two different triangles, or none. It does NOT guarantee congruence — this is the "ambiguous case."

In quadrilateral \(ABCD\), diagonals \(\overline{AC}\) and \(\overline{BD}\) bisect each other at \(M\).
Which congruence postulate directly proves \(\triangle AMB \cong \triangle CMD\)?

AREA OF REGULAR POLYGON — "A = ½·perimeter·apothem · apothem = side/(2·tan(π/n))"

G·09 HARD ★★★★★

A regular hexagon has a side length of \(8\) cm.
Find its exact area. (Do not use a calculator — leave in simplest radical form.)

TRIGONOMETRY (SOH-CAH-TOA) — "sin=opp/hyp · cos=adj/hyp · tan=opp/adj · Law of Cosines for non-right"

G·10 HARD ★★★★★

📖 Law of Cosines \(c^2 = a^2 + b^2 - 2ab\cos C\) — use when you know SAS or SSS.

In \(\triangle PQR\), \(PQ = 7\), \(QR = 9\), and \(\angle Q = 120°\).
Find \(PR\) to the nearest integer.