Algebra 2 & Geometry · Practice Problems

A · 01 Quadratic Equations

DISCRIMINANT b²−4ac → >0 two real · =0 one · <0 no real

Which equation has no real solutions?

💡 Tip
Calculate \(b^2 - 4ac\) for each. If the result is negative, no real roots exist.

📖 Explanation

For (C): \(b^2 - 4ac = 1 - 8 = -7 < 0\) → no real solutions.
(A): \(25-24=1>0\) → 2 real roots (x=2, x=3).
(B): \(0+16=16>0\) → 2 real roots (x=±2).
(D): \(9-0=9>0\) → 2 real roots (x=0, x=3/2).

A · 02 Vertex Form

VERTEX FORM y = a(x−h)²+k → vertex (h, k)

The parabola \(y = 2(x-3)^2 + 5\) has its vertex at:

📖 Explanation

In \(y = a(x-h)^2+k\), the vertex is \((h,\ k)\).
Here \(h=3,\ k=5\), so vertex = (3, 5).
⚠️ Common mistake: students write (−3, 5) because they forget the sign flips.

A · 03 Logarithms

LOG POWER RULE log(aⁿ) = n·log(a)

Simplify: \(\log_2 8 + \log_2 4\)

💡 Tip
Use the product rule: \(\log_b m + \log_b n = \log_b(mn)\).
Or compute each separately: \(\log_2 8 = 3\) because \(2^3=8\).

📖 Explanation

\(\log_2 8 = 3\) (since \(2^3=8\))
\(\log_2 4 = 2\) (since \(2^2=4\))
Sum = \(3 + 2 = \mathbf{5}\).
Or: \(\log_2(8 \times 4) = \log_2 32 = 5\).

A · 04 Polynomial Division / Remainder Theorem

REMAINDER THEOREM f(a) = remainder when f(x) ÷ (x−a)

When \(f(x) = x^3 - 2x + 5\) is divided by \((x - 2)\), the remainder is:

📖 Explanation

Plug \(x=2\) into \(f(x)\):
\(f(2) = 2^3 - 2(2) + 5 = 8 - 4 + 5 = \mathbf{9}\).
By the Remainder Theorem, the remainder = 9.

A · 05 Complex Numbers

i² = −1 i³ = −i i⁴ = 1 → cycle of 4

What is \((3 + 2i)(1 - i)\)?

💡 Tip
FOIL it out, then replace \(i^2\) with \(-1\).

📖 Explanation

\((3+2i)(1-i) = 3 - 3i + 2i - 2i^2\)
\(= 3 - i - 2(-1) = 3 - i + 2 = \mathbf{5 - i}\).

A · 06 Exponential Functions

GROWTH/DECAY y = a·bˣ → b>1 growth · 0<b<1 decay

Which function represents exponential decay?

📖 Explanation

In \(y = a \cdot b^x\), decay requires 0 < b < 1.
(B) has base 0.4, which is between 0 and 1 → decay.
(D) is a reflection (negative a), not decay in the standard sense.
(A) base=2 > 1 → growth. (C) same issue.

A · 07 Arithmetic Sequences

Aₙ = a₁ + (n−1)d d = common difference

The 15th term of the arithmetic sequence \(3, 7, 11, 15, \ldots\) is:

📖 Explanation

\(a_1=3,\ d=4\)
\(a_{15} = 3 + (15-1)\cdot 4 = 3 + 56 = \mathbf{59}\).
⚠️ Remember: multiply by (n−1), not n.

A · 08 Rational Functions

VERTICAL ASYMPTOTE set denominator = 0 (after canceling)

What is the vertical asymptote of \(\displaystyle f(x) = \frac{x+1}{x^2 - 9}\)?

💡 Tip
Factor denominator: \(x^2 - 9 = (x-3)(x+3)\). Set each factor = 0.

📖 Explanation

\(x^2-9 = (x-3)(x+3) = 0 \Rightarrow x=3 \text{ or } x=-3\).
Neither zero cancels with the numerator (\(x+1\)), so both are vertical asymptotes.
Answer: x = 3 and x = −3.

A · 09 Inverse Functions

INVERSE swap x & y, solve for y → write f⁻¹(x)

If \(f(x) = 2x - 6\), then \(f^{-1}(x) =\)

📖 Explanation

Step 1: Replace \(f(x)\) with \(y\): \(y = 2x-6\)
Step 2: Swap \(x\) and \(y\): \(x = 2y-6\)
Step 3: Solve for \(y\): \(y = \dfrac{x+6}{2}\)
So \(f^{-1}(x) = \mathbf{\dfrac{x+6}{2}}\).

A · 10 Systems of Equations

SUBSTITUTION isolate one variable → plug into other equation

Solve the system: \(y = x^2 - 4\) and \(y = x - 2\). The solution(s) are:

💡 Tip
Set them equal: \(x^2-4 = x-2\). Rearrange and factor.

📖 Explanation

\(x^2-4 = x-2\)
\(x^2-x-2 = 0\)
\((x-2)(x+1)=0 \Rightarrow x=2 \text{ or } x=-1\)
For \(x=2\): \(y=0\). For \(x=-1\): \(y=-3\).
Solutions: (2, 0) and (−1, −3).

G · 01 Triangle Angle Sum

TRIANGLE SUM three angles always add to 180°

In a triangle, two angles measure 47° and 83°. The third angle is:

📖 Explanation

Sum of angles = 180°
Third angle = \(180 - 47 - 83 = \mathbf{50°}\).

G · 02 Pythagorean Theorem

a²+b²=c² c is always the hypotenuse (longest side)

A right triangle has legs of length 6 and 8. What is the hypotenuse?

📖 Explanation

\(c = \sqrt{6^2 + 8^2} = \sqrt{36+64} = \sqrt{100} = \mathbf{10}\).
This is the famous 3-4-5 triple, scaled by 2: 6-8-10.

G · 03 Circle — Arc & Central Angle

CENTRAL ANGLE = intercepted arc (same degree measure)

A central angle of 120° intercepts an arc. If the circle has radius 9, what is the arc length?

💡 Formula
Arc length \(= \dfrac{\theta}{360} \times 2\pi r\)

📖 Explanation

Arc length \(= \dfrac{120}{360} \times 2\pi(9) = \dfrac{1}{3} \times 18\pi = \mathbf{6\pi}\).

G · 04 Similar Triangles

AA SIMILARITY two pairs of equal angles → triangles are similar

Triangles ABC and DEF are similar with ratio 3:5. If \(AB = 12\), then \(DE =\)

📖 Explanation

Ratio ABC : DEF = 3 : 5.
\(\dfrac{AB}{DE} = \dfrac{3}{5} \Rightarrow \dfrac{12}{DE} = \dfrac{3}{5}\)
\(DE = \dfrac{12 \times 5}{3} = \mathbf{20}\).

G · 05 Coordinate Geometry — Midpoint

MIDPOINT M = ( (x₁+x₂)/2 , (y₁+y₂)/2 )

Find the midpoint of the segment with endpoints \((-2,\ 6)\) and \((8,\ -4)\).

📖 Explanation

\(M_x = \dfrac{-2+8}{2} = 3\)
\(M_y = \dfrac{6+(-4)}{2} = 1\)
Midpoint = (3, 1).

G · 06 Parallel Lines & Transversals

ALTERNATE INTERIOR angles are equal when lines are parallel

Two parallel lines are cut by a transversal. One alternate interior angle is \((3x+10)°\) and the other is \((5x-14)°\). Find \(x\).

📖 Explanation

Alternate interior angles are equal:
\(3x+10 = 5x-14\)
\(24 = 2x\)
\(x = \mathbf{12}\).

G · 07 Volume — Cone

CONE VOLUME V = ⅓πr²h (one-third of cylinder)

A cone has radius 3 and height 7. Its volume is:

📖 Explanation

\(V = \dfrac{1}{3}\pi(3)^2(7) = \dfrac{1}{3}\pi \cdot 9 \cdot 7 = \dfrac{63\pi}{3} = \mathbf{21\pi}\).
⚠️ Don't forget the \(\dfrac{1}{3}\)!

G · 08 Inscribed Angle Theorem

INSCRIBED ANGLE = ½ × intercepted arc

An inscribed angle in a circle intercepts an arc of 140°. What is the measure of the inscribed angle?

📖 Explanation

Inscribed Angle = \(\dfrac{1}{2}\) × arc = \(\dfrac{1}{2} \times 140° = \mathbf{70°}\).
⚠️ Inscribed angle is HALF the arc. Central angle equals the arc.

G · 09 Transformations — Reflection

REFLECT over y-axis → (x, y) becomes (−x, y)

Point \(P(4, -3)\) is reflected across the y-axis. What are its new coordinates?

📖 Explanation

Reflection over y-axis: only the x-coordinate changes sign.
\((4, -3) \rightarrow \mathbf{(-4, -3)}\).
Over x-axis → y changes sign. Over y-axis → x changes sign.

G · 10 Special Right Triangles

30-60-90 sides: x · x√3 · 2x | 45-45-90 sides: x · x · x√2

In a 30-60-90 triangle, the hypotenuse is 16. What is the length of the shorter leg?

💡 Tip
In a 30-60-90 triangle: shorter leg = hypotenuse ÷ 2.

📖 Explanation

In 30-60-90: if hypotenuse \(= 2x\), then shorter leg \(= x\) and longer leg \(= x\sqrt{3}\).
Hypotenuse = 16 → \(2x=16\) → \(x=\mathbf{8}\).
Longer leg = \(8\sqrt{3}\). Shorter leg = 8.