Algebra 2
10 Problems
Quick Memory Key
DISCRIMINANT TELLS ALL — If \(b^2-4ac > 0\): 2 real roots · \(= 0\): 1 root · \(< 0\): no real roots
📖 Example
Solve \(x^2 - 5x + 6 = 0\).
Using \(x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\) with \(a=1,b=-5,c=6\):
\(x = \dfrac{5 \pm \sqrt{25-24}}{2} = \dfrac{5 \pm 1}{2}\) → \(x = 3\) or \(x = 2\)
Using \(x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\) with \(a=1,b=-5,c=6\):
\(x = \dfrac{5 \pm \sqrt{25-24}}{2} = \dfrac{5 \pm 1}{2}\) → \(x = 3\) or \(x = 2\)
How many real solutions does \(2x^2 - 3x + 5 = 0\) have?
💡 Explanation
Compute the discriminant: \(b^2 - 4ac = (-3)^2 - 4(2)(5) = 9 - 40 = -31\).
Since \(-31 < 0\), the square root of a negative number is not real → No real solutions.
Since \(-31 < 0\), the square root of a negative number is not real → No real solutions.
Quick Memory Key
HALF · SQUARE · BALANCE — Take half the \(b\)-coefficient, square it, ADD to both sides
📖 Example
Complete the square: \(x^2 + 6x = 7\)
Half of 6 is 3 → add \(3^2 = 9\) to both sides:
\(x^2 + 6x + 9 = 16\) → \((x+3)^2 = 16\) → \(x = 1\) or \(x = -7\)
Half of 6 is 3 → add \(3^2 = 9\) to both sides:
\(x^2 + 6x + 9 = 16\) → \((x+3)^2 = 16\) → \(x = 1\) or \(x = -7\)
Complete the square to rewrite \(x^2 - 8x + 7 = 0\) in vertex form. What is the vertex of the related parabola?
💡 Explanation
\(x^2 - 8x + 7 = 0\) → \(x^2 - 8x = -7\)
Half of \(-8\) is \(-4\); \((-4)^2 = 16\). Add 16 to both sides:
\((x-4)^2 = 9\)
Vertex form: \(y = (x-4)^2 - 9\) → Vertex = \(\mathbf{(4,\ -9)}\).
Half of \(-8\) is \(-4\); \((-4)^2 = 16\). Add 16 to both sides:
\((x-4)^2 = 9\)
Vertex form: \(y = (x-4)^2 - 9\) → Vertex = \(\mathbf{(4,\ -9)}\).
Quick Memory Key
SAME BASE → SET EXPONENTS EQUAL — If \(a^m = a^n\), then \(m = n\) (as long as \(a > 0, a \neq 1\))
📖 Example
Solve \(4^x = 8\)
Rewrite with same base: \(2^{2x} = 2^3\) → \(2x = 3\) → \(x = \dfrac{3}{2}\)
Rewrite with same base: \(2^{2x} = 2^3\) → \(2x = 3\) → \(x = \dfrac{3}{2}\)
Solve for \(x\): \(9^x = 27\)
💡 Explanation
Rewrite both sides as powers of 3:
\(9^x = (3^2)^x = 3^{2x}\) and \(27 = 3^3\)
So \(3^{2x} = 3^3\) → \(2x = 3\) → \(\boxed{x = \dfrac{3}{2}}\)
\(9^x = (3^2)^x = 3^{2x}\) and \(27 = 3^3\)
So \(3^{2x} = 3^3\) → \(2x = 3\) → \(\boxed{x = \dfrac{3}{2}}\)
Quick Memory Key
LOG = EXPONENT — \(\log_b x = y\) means \(b^y = x\). Log asks: "What power?"
📖 Example
\(\log_2 32 = ?\)
"2 to the what power equals 32?" → \(2^5 = 32\) → Answer: \(5\)
"2 to the what power equals 32?" → \(2^5 = 32\) → Answer: \(5\)
Evaluate: \(\log_3 \dfrac{1}{81}\)
💡 Explanation
Ask: "3 to the what power equals \(\tfrac{1}{81}\)?"
\(\dfrac{1}{81} = \dfrac{1}{3^4} = 3^{-4}\)
So \(\log_3 3^{-4} = \boxed{-4}\)
\(\dfrac{1}{81} = \dfrac{1}{3^4} = 3^{-4}\)
So \(\log_3 3^{-4} = \boxed{-4}\)
Quick Memory Key
VA: DENOMINATOR = 0 · HA: COMPARE DEGREES — Vertical asymptote where denom = 0; Horizontal asymptote depends on degree comparison
📖 Example
\(f(x) = \dfrac{2x}{x-3}\): VA at \(x=3\); degrees are equal so HA = \(\dfrac{2}{1} = 2\)
What is the horizontal asymptote of \(f(x) = \dfrac{3x^2 - 1}{x^2 + 2}\)?
💡 Explanation
Numerator degree = 2, denominator degree = 2 → equal degrees.
Horizontal asymptote = ratio of leading coefficients = \(\dfrac{3}{1} = \boxed{3}\), so \(y = 3\).
Horizontal asymptote = ratio of leading coefficients = \(\dfrac{3}{1} = \boxed{3}\), so \(y = 3\).
Quick Memory Key
DIFFERENCE OF CUBES: a³−b³ = (a−b)(a²+ab+b²) — middle sign of trinomial is always POSITIVE
📖 Example
Factor \(x^3 - 8\): Here \(a = x, b = 2\)
\((x-2)(x^2 + 2x + 4)\)
\((x-2)(x^2 + 2x + 4)\)
Which is the correct factored form of \(8x^3 - 27\)?
💡 Explanation
\(8x^3 - 27 = (2x)^3 - 3^3\). Using \(a^3 - b^3 = (a-b)(a^2+ab+b^2)\):
\(a = 2x,\ b = 3\) →
\(\boxed{(2x-3)((2x)^2 + (2x)(3) + 3^2)} = (2x-3)(4x^2+6x+9)\)
\(a = 2x,\ b = 3\) →
\(\boxed{(2x-3)((2x)^2 + (2x)(3) + 3^2)} = (2x-3)(4x^2+6x+9)\)
Quick Memory Key
SUBSTITUTION → PLUG IN → SOLVE BACK — Isolate one variable, substitute, find the other, then substitute back
📖 Example
\(y = 2x\) and \(y = x + 3\) → Sub: \(2x = x + 3\) → \(x = 3, y = 6\)
Solve the system: \(y = x^2 - 2\) and \(y = 2x + 1\). How many intersection points are there?
💡 Explanation
Set equal: \(x^2 - 2 = 2x + 1\) → \(x^2 - 2x - 3 = 0\) → \((x-3)(x+1) = 0\)
\(x = 3\) or \(x = -1\) → 2 intersection points: \((3, 7)\) and \((-1, -1)\).
\(x = 3\) or \(x = -1\) → 2 intersection points: \((3, 7)\) and \((-1, -1)\).
Quick Memory Key
i² = −1 · i³ = −i · i⁴ = 1 · CYCLE OF 4 — Powers of \(i\) repeat every 4 steps
📖 Example
\(i^{10} = i^{8+2} = (i^4)^2 \cdot i^2 = 1 \cdot (-1) = -1\)
Simplify: \((3 + 2i)(1 - 4i)\)
💡 Explanation
FOIL: \((3)(1) + (3)(-4i) + (2i)(1) + (2i)(-4i)\)
\(= 3 - 12i + 2i - 8i^2\)
\(= 3 - 10i - 8(-1)\)
\(= 3 - 10i + 8 = \boxed{11 - 10i}\)
\(= 3 - 12i + 2i - 8i^2\)
\(= 3 - 10i - 8(-1)\)
\(= 3 - 10i + 8 = \boxed{11 - 10i}\)
Quick Memory Key
SWAP x AND y → SOLVE FOR y — To find inverse, exchange \(x\) and \(y\), then isolate \(y\)
📖 Example
\(f(x) = 2x - 4\) → swap: \(x = 2y - 4\) → \(y = \dfrac{x+4}{2}\) → \(f^{-1}(x) = \dfrac{x+4}{2}\)
Find \(f^{-1}(x)\) for \(f(x) = \dfrac{x+3}{5}\)
💡 Explanation
Let \(y = \dfrac{x+3}{5}\). Swap \(x\) and \(y\):
\(x = \dfrac{y+3}{5}\) → \(5x = y + 3\) → \(y = 5x - 3\)
So \(\boxed{f^{-1}(x) = 5x - 3}\)
\(x = \dfrac{y+3}{5}\) → \(5x = y + 3\) → \(y = 5x - 3\)
So \(\boxed{f^{-1}(x) = 5x - 3}\)
Quick Memory Key
aₙ = a₁ + (n−1)d — Start value + (how many gaps) × (gap size). Don't forget to subtract 1!
📖 Example
Sequence: 3, 7, 11, 15… → \(a_1 = 3, d = 4\)
\(a_{10} = 3 + 9(4) = 39\)
\(a_{10} = 3 + 9(4) = 39\)
In the arithmetic sequence \(5, 11, 17, 23, \ldots\), what is the 50th term?
💡 Explanation
\(a_1 = 5,\ d = 11 - 5 = 6\)
\(a_{50} = 5 + (50-1)(6) = 5 + 49 \times 6 = 5 + 294 = \boxed{299}\)
\(a_{50} = 5 + (50-1)(6) = 5 + 49 \times 6 = 5 + 294 = \boxed{299}\)
Geometry
Geometry
10 Problems
Quick Memory Key
SSS · SAS · ASA · AAS · HL — These 5 prove congruence. AAA and SSA do NOT work!
📖 Example
Two triangles share a common side. We know two sides and the included angle of each are equal → use SAS to prove congruence.
Two triangles have two pairs of equal sides and the angle between those sides is equal. Which congruence theorem applies?
💡 Explanation
Two equal Sides with the equal angle Angle between them = Side-Angle-Side (SAS).
The key word is "between" — if the angle were not between the two sides, SAS would not apply.
The key word is "between" — if the angle were not between the two sides, SAS would not apply.
Quick Memory Key
LEG² + LEG² = HYPOTENUSE² — Hypotenuse is ALWAYS the longest side, opposite the right angle
📖 Example
Legs: 3 and 4 → \(3^2 + 4^2 = 9 + 16 = 25\) → hypotenuse = \(\sqrt{25} = 5\)
A right triangle has legs of length \(5\) and \(12\). What is the length of the hypotenuse?
💡 Explanation
\(c^2 = 5^2 + 12^2 = 25 + 144 = 169\)
\(c = \sqrt{169} = \boxed{13}\)
(5, 12, 13) is a classic Pythagorean triple!
\(c = \sqrt{169} = \boxed{13}\)
(5, 12, 13) is a classic Pythagorean triple!
Quick Memory Key
SECTOR = (θ/360°) × πr² — Fraction of the full circle. Arc length = (θ/360°) × 2πr
📖 Example
Circle r = 6, central angle = 90°
Sector area = \(\dfrac{90}{360} \times \pi(6)^2 = \dfrac{1}{4} \times 36\pi = 9\pi\)
Sector area = \(\dfrac{90}{360} \times \pi(6)^2 = \dfrac{1}{4} \times 36\pi = 9\pi\)
A circle has radius \(r = 10\). A sector has a central angle of \(72°\). What is the area of the sector?
💡 Explanation
\(\text{Sector area} = \dfrac{72}{360} \times \pi (10)^2 = \dfrac{1}{5} \times 100\pi = \boxed{20\pi}\)
Quick Memory Key
SAME ANGLES → SIDES PROPORTIONAL — Set up ratios: corresponding sides / corresponding sides = constant ratio
📖 Example
Triangles with sides 3,4,5 and 6,8,? → ratio = 2:1 → missing side = 10
Two similar triangles have corresponding sides in the ratio \(3:5\). If the area of the smaller triangle is \(27\ \text{cm}^2\), what is the area of the larger triangle?
💡 Explanation
Area scales as the SQUARE of the side ratio.
Side ratio = \(\dfrac{3}{5}\) → Area ratio = \(\left(\dfrac{3}{5}\right)^2 = \dfrac{9}{25}\)
\(\dfrac{27}{\text{Area}_{\text{large}}} = \dfrac{9}{25}\) → \(\text{Area}_{\text{large}} = 27 \times \dfrac{25}{9} = \boxed{75\ \text{cm}^2}\)
Side ratio = \(\dfrac{3}{5}\) → Area ratio = \(\left(\dfrac{3}{5}\right)^2 = \dfrac{9}{25}\)
\(\dfrac{27}{\text{Area}_{\text{large}}} = \dfrac{9}{25}\) → \(\text{Area}_{\text{large}} = 27 \times \dfrac{25}{9} = \boxed{75\ \text{cm}^2}\)
Quick Memory Key
Z = ALTERNATE (equal) · F = CORRESPONDING (equal) · C = CO-INTERIOR (add to 180°)
📖 Example
Two parallel lines cut by a transversal: co-interior angles sum to 180°. If one is 110°, the other is 70°.
Two parallel lines are cut by a transversal. One co-interior (same-side interior) angle is \(65°\). What is the measure of the other co-interior angle?
💡 Explanation
Co-interior angles (same-side interior) are supplementary — they add up to \(180°\).
\(180° - 65° = \boxed{115°}\)
Don't confuse with alternate interior angles, which are equal!
\(180° - 65° = \boxed{115°}\)
Don't confuse with alternate interior angles, which are equal!
Quick Memory Key
CONE = ⅓ × CYLINDER — Same base and height: a cone holds exactly 1/3 of a cylinder's volume
📖 Example
Cone: \(r=3, h=4\) → \(V = \dfrac{1}{3}\pi(3)^2(4) = 12\pi\)
A cone and a cylinder have the same base radius \(r = 6\) and height \(h = 9\). The volume of the cone is what fraction of the cylinder's volume?
💡 Explanation
\(V_{\text{cylinder}} = \pi r^2 h = 324\pi\)
\(V_{\text{cone}} = \dfrac{1}{3}\pi r^2 h = 108\pi\)
\(\dfrac{108\pi}{324\pi} = \boxed{\dfrac{1}{3}}\)
This is always true: a cone is always \(\tfrac{1}{3}\) of the cylinder with same base and height.
\(V_{\text{cone}} = \dfrac{1}{3}\pi r^2 h = 108\pi\)
\(\dfrac{108\pi}{324\pi} = \boxed{\dfrac{1}{3}}\)
This is always true: a cone is always \(\tfrac{1}{3}\) of the cylinder with same base and height.
Quick Memory Key
INSCRIBED = HALF THE CENTRAL — Inscribed angle = ½ × intercepted arc. Angle in semicircle = 90°
📖 Example
Central angle = 80° → Inscribed angle intercepting same arc = 40°
An inscribed angle in a circle intercepts an arc of \(140°\). What is the measure of the inscribed angle?
💡 Explanation
Inscribed Angle Theorem: inscribed angle = \(\dfrac{1}{2} \times\) intercepted arc
\(= \dfrac{1}{2} \times 140° = \boxed{70°}\)
\(= \dfrac{1}{2} \times 140° = \boxed{70°}\)
Quick Memory Key
45-45-90: x · x · x√2 and 30-60-90: x · x√3 · 2x — Memorize these ratios!
📖 Example
30-60-90 with shortest leg = 5:
Other leg = \(5\sqrt{3}\), hypotenuse = \(10\)
Other leg = \(5\sqrt{3}\), hypotenuse = \(10\)
In a 45-45-90 triangle, the hypotenuse is \(8\sqrt{2}\). What is the length of each leg?
💡 Explanation
In a 45-45-90 triangle: hypotenuse = leg × \(\sqrt{2}\)
So: leg = \(\dfrac{8\sqrt{2}}{\sqrt{2}} = \boxed{8}\)
So: leg = \(\dfrac{8\sqrt{2}}{\sqrt{2}} = \boxed{8}\)
Quick Memory Key
MIDPOINT = AVERAGE the coordinates — \(M = \left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right)\)
📖 Example
Midpoint of \((2,4)\) and \((8,10)\):
\(M = \left(\dfrac{2+8}{2},\dfrac{4+10}{2}\right) = (5, 7)\)
\(M = \left(\dfrac{2+8}{2},\dfrac{4+10}{2}\right) = (5, 7)\)
Point \(M\) is the midpoint of \(\overline{AB}\). If \(A = (-3,\ 7)\) and \(M = (2,\ 1)\), what are the coordinates of point \(B\)?
💡 Explanation
Midpoint formula: \(M = \left(\dfrac{x_A + x_B}{2},\ \dfrac{y_A+y_B}{2}\right)\)
\(2 = \dfrac{-3 + x_B}{2} \Rightarrow x_B = 7\)
\(1 = \dfrac{7 + y_B}{2} \Rightarrow y_B = -5\)
So \(B = \boxed{(7,\ -5)}\)
\(2 = \dfrac{-3 + x_B}{2} \Rightarrow x_B = 7\)
\(1 = \dfrac{7 + y_B}{2} \Rightarrow y_B = -5\)
So \(B = \boxed{(7,\ -5)}\)
Quick Memory Key
SA = 4πr² and V = (4/3)πr³ — Sphere has 4 great circles worth of surface. "4 pizzas cover a ball!"
📖 Example
Sphere \(r = 3\): \(SA = 4\pi(9) = 36\pi\) and \(V = \dfrac{4}{3}\pi(27) = 36\pi\)
A sphere has a surface area of \(100\pi\ \text{cm}^2\). What is the radius of the sphere?
💡 Explanation
\(4\pi r^2 = 100\pi\)
\(r^2 = \dfrac{100\pi}{4\pi} = 25\)
\(r = \sqrt{25} = \boxed{5\ \text{cm}}\)
\(r^2 = \dfrac{100\pi}{4\pi} = 25\)
\(r = \sqrt{25} = \boxed{5\ \text{cm}}\)