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ALGEBRA — COMPLETE SAVED LEARNINGS
ax2 + bx + c = 0
The sign is included in b.
Example:
2x2 - 5x + 3 = 0
means:
a = 2
b = -5
c = 3
Important: Don't treat b as 5. Here, b = -5.
D > 0 → 2 distinct real roots
D = 0 → 1 repeated real root
D < 0 → no real roots
ax2 + bx + c
Vertex occurs at:
x = -b/(2a)
a > 0 → minimum
a < 0 → maximum
OPTIMIZATION / EQUALIZATION
x + y = S
Then xy is maximized when:
x = y = S/2
Maximum:
xy = S2/4
Recognition: Fixed sum + maximize product → equalize.
xy = P
Then:
x + y ≥ 2√P
Minimum occurs at:
x = y = √P
Recognition: Fixed product + minimize sum → equalize.
a + b ≥ 2√(ab)
Equality occurs when:
a = b
Important disguised form:
x + k/x ≥ 2√k
for x > 0.
Example:
x + 16/x ≥ 8
Minimum = 8 at x = 4.
x + y = S
For:
x2 + y2
Minimum → equalize.
Example:
x + y = 30
Minimum:
152 + 152
Maximum → push apart, subject to the allowed domain/constraints.
Example for positive integers:
292 + 12
Recognition: Squares reward extremes.
PARAMETER INEQUALITIES
"Largest/smallest value of k such that an inequality is possible/always true"
The boundary generally occurs at equality.
Example:
x2 - 6x + 10 ≥ k
Complete the square:
(x - 3)2 + 1 ≥ k
Minimum LHS = 1
Therefore, largest possible k = 1.
Recognition:
"Largest constant such that Expression ≥ k"
→ Find the MINIMUM of the expression.
"Smallest constant such that Expression ≤ k"
→ Find the MAXIMUM of the expression.
ABSOLUTE VALUES
Therefore:
|x - a||x + a|
becomes:
|x2 - a2|
Example:
|x - 5||x + 5| = 24
becomes:
|x2 - 25| = 24
So:
x2 - 25 = ±24
giving:
x = ±7, ±1
Recognition:
|x - a||x + a| → |x2 - a2|
means x is equally distant from a and b.
Therefore:
x = (a + b)/2
|x - a1| + |x - a2| + ... + |x - aₙ|
The minimum occurs at a median of the ai's.
ROOTS
Therefore:
√25 = 5
NOT:
±5
But:
x2 = 25
gives:
x = ±5
Recognition:
Root already exists → principal/nonnegative value.
You solve x2 = a → consider ±.
Always check the solutions in the ORIGINAL equation.
Squaring can create invalid solutions.
√(x + a) + √(x + b) = c
Don't square the whole expression immediately.
Instead:
Isolate one root → square → isolate remaining root → square again.
Then verify solutions.
EXPONENT ALGEBRA
an/bn = (a/b)n
Example:
85 × 1255
= (8 × 125)5
= 10005
Recognition: Same exponent → combine bases.
ax = t
and asked for something whose base is ak:
(ak)x = (ax)k = tk
Example:
2x = 5
Find 8x.
8x
= (23)x
= (2x)3
= 53
= 125
Recognition: Don't solve for x. Rewrite the base.
a2x and ax
Recognize:
a2x = (ax)2
Let:
y = ax
Example:
22x - 5(2x) + 4 = 0
becomes:
y2 - 5y + 4 = 0
a > 1
Then:
ax > ay ↔ x > y
Order is preserved.
0 < a < 1
Then:
ax > ay ↔ x < y
Direction reverses.
Example:
(1/2)^(2x - 1) > (1/2)5
Therefore:
2x - 1 < 5
x < 3
a0 = 1, for a ≠ 0
0n = 0, for n > 0
00 = undefined/indeterminate
Also:
ax = ay → x = y
requires appropriate base conditions, typically:
a > 0 and a ≠ 1
If a = 1, the exponents tell you nothing.
COMPARING POWERS / NUMBERS
0 < x < 1
x2 < x < √x < 1 < 1/x
So:
• Squaring makes x smaller.
• Square root makes x larger but still < 1.
• Reciprocal jumps above 1.
x > 1
1/x < 1 < √x < x < x2
So:
• Squaring makes x larger.
• 2x > x.
• Reciprocal jumps below 1.
Key saved idea:
Whether x > 1 or 0 < x < 1:
Taking the reciprocal jumps to the opposite side of 1.
Exception: x = 1.
Greatest negative number → remove negatives → choose SMALLEST magnitude.
Least negative number → remove negatives → choose LARGEST magnitude.
Example:
-2, -5, -9
Greatest = -2
Least = -9
a−n = 1/an
Then compare.
If:
0 < a < b
Then:
a2 < b2
Order preserved.
Both negative:
If:
a < b < 0
Then:
a2 > b2
Order reverses.
Mixed signs:
No fixed comparison rule from squaring alone.
Recognition: Check signs before squaring to compare.
FRACTION / ROOT ALGEBRA COMPARISON
√a / b = √(a/b2)
a / √b = √(a2/b)
Then compare the radicands.
Since √x is increasing for x ≥ 0:
Larger radicand → larger square root.
FRACTION MOVEMENT
a/b
Adding k > 0:
(a + k)/(b + k)
moves the fraction toward 1.
If:
a/b < 1
Then:
(a + k)/(b + k) > a/b
If:
a/b > 1
Then:
(a + k)/(b + k) < a/b
Memory:
Adding the same positive amount to the top and bottom pulls the fraction toward 1.
ad > bc → a/b > c/d
ad < bc → a/b < c/d
Increasing the numerator → increases the fraction.
Increasing the denominator → decreases the fraction.
So, all else equal:
Larger numerator + smaller denominator → larger fraction.
This applies whether the fractions are below or above 1.
CUSTOM / INTEGER FUNCTION
Examples:
⌊3.8⌋ = 3
⌊-2.3⌋ = -3
Important: For negative numbers, ⌊-2.3⌋ = -3, NOT -2.
MASTER ALGEBRA RECOGNITION
When you see:
• Fixed sum + maximum product → equalize.
• Fixed product + minimum sum → equalize.
• x + k/x, with x > 0 → AM-GM.
• Largest k with Expression ≥ k → find the expression's minimum.
• |x - a||x + a| → |x2 - a2|.
• Sum of many absolute values → median.
• Square root equation → remember the root is nonnegative; verify after squaring.
• Same exponent → combine bases.
• Repeated ax → substitute.
• Given ax, asked for related base → rewrite the base; don't solve x.
• Base > 1 → exponent inequality direction stays the same.
• 0 < base < 1 → exponent inequality direction reverses.
• 0 < x < 1 → x2 < x < √x < 1 < 1/x.
• x > 1 → 1/x < 1 < √x < x < x2.
• Compare by squaring → check signs first.
• Roots + fractions → put everything under one square root.
• Same amount added to numerator and denominator → fraction moves toward 1.
• Floor of a negative decimal → move DOWN to the next smaller integer.
- QUADRATIC — STANDARD FORM
ax2 + bx + c = 0
The sign is included in b.
Example:
2x2 - 5x + 3 = 0
means:
a = 2
b = -5
c = 3
Important: Don't treat b as 5. Here, b = -5.
- DISCRIMINANT — NUMBER OF REAL SOLUTIONS
D > 0 → 2 distinct real roots
D = 0 → 1 repeated real root
D < 0 → no real roots
- QUADRATIC VERTEX
ax2 + bx + c
Vertex occurs at:
x = -b/(2a)
a > 0 → minimum
a < 0 → maximum
OPTIMIZATION / EQUALIZATION
- FIXED SUM → PRODUCT MAXIMIZED WHEN EQUAL ⭐
x + y = S
Then xy is maximized when:
x = y = S/2
Maximum:
xy = S2/4
Recognition: Fixed sum + maximize product → equalize.
- FIXED PRODUCT → SUM MINIMIZED WHEN EQUAL ⭐
xy = P
Then:
x + y ≥ 2√P
Minimum occurs at:
x = y = √P
Recognition: Fixed product + minimize sum → equalize.
- AM-GM ⭐
a + b ≥ 2√(ab)
Equality occurs when:
a = b
Important disguised form:
x + k/x ≥ 2√k
for x > 0.
Example:
x + 16/x ≥ 8
Minimum = 8 at x = 4.
- FIXED SUM → SUM OF SQUARES
x + y = S
For:
x2 + y2
Minimum → equalize.
Example:
x + y = 30
Minimum:
152 + 152
Maximum → push apart, subject to the allowed domain/constraints.
Example for positive integers:
292 + 12
Recognition: Squares reward extremes.
PARAMETER INEQUALITIES
- BOUNDARY = EQUALITY ⭐
"Largest/smallest value of k such that an inequality is possible/always true"
The boundary generally occurs at equality.
Example:
x2 - 6x + 10 ≥ k
Complete the square:
(x - 3)2 + 1 ≥ k
Minimum LHS = 1
Therefore, largest possible k = 1.
Recognition:
"Largest constant such that Expression ≥ k"
→ Find the MINIMUM of the expression.
"Smallest constant such that Expression ≤ k"
→ Find the MAXIMUM of the expression.
ABSOLUTE VALUES
- ABSOLUTE VALUE PRODUCT IDENTITY ⭐
Therefore:
|x - a||x + a|
becomes:
|x2 - a2|
Example:
|x - 5||x + 5| = 24
becomes:
|x2 - 25| = 24
So:
x2 - 25 = ±24
giving:
x = ±7, ±1
Recognition:
|x - a||x + a| → |x2 - a2|
- SAME DISTANCE IDENTITY
means x is equally distant from a and b.
Therefore:
x = (a + b)/2
- SUM OF MANY ABSOLUTE VALUES — MEDIAN RULE
|x - a1| + |x - a2| + ... + |x - aₙ|
The minimum occurs at a median of the ai's.
ROOTS
- SQUARE ROOT IS ALWAYS NONNEGATIVE ⭐
Therefore:
√25 = 5
NOT:
±5
But:
x2 = 25
gives:
x = ±5
Recognition:
Root already exists → principal/nonnegative value.
You solve x2 = a → consider ±.
- SQUARING CAN INTRODUCE EXTRANEOUS ROOTS ⭐
Always check the solutions in the ORIGINAL equation.
Squaring can create invalid solutions.
- EQUATIONS WITH TWO SQUARE ROOTS
√(x + a) + √(x + b) = c
Don't square the whole expression immediately.
Instead:
Isolate one root → square → isolate remaining root → square again.
Then verify solutions.
EXPONENT ALGEBRA
- SAME EXPONENT → COMBINE BASES ⭐
an/bn = (a/b)n
Example:
85 × 1255
= (8 × 125)5
= 10005
Recognition: Same exponent → combine bases.
- REWRITE BASE INSTEAD OF SOLVING EXPONENT ⭐
ax = t
and asked for something whose base is ak:
(ak)x = (ax)k = tk
Example:
2x = 5
Find 8x.
8x
= (23)x
= (2x)3
= 53
= 125
Recognition: Don't solve for x. Rewrite the base.
- REPEATED ax → SUBSTITUTE ⭐
a2x and ax
Recognize:
a2x = (ax)2
Let:
y = ax
Example:
22x - 5(2x) + 4 = 0
becomes:
y2 - 5y + 4 = 0
- EXPONENTIAL INEQUALITIES — BASE > 1
a > 1
Then:
ax > ay ↔ x > y
Order is preserved.
- EXPONENTIAL INEQUALITIES — BASE BETWEEN 0 AND 1
0 < a < 1
Then:
ax > ay ↔ x < y
Direction reverses.
Example:
(1/2)^(2x - 1) > (1/2)5
Therefore:
2x - 1 < 5
x < 3
- ZERO / ONE EXPONENT EDGE CASES ⭐
a0 = 1, for a ≠ 0
0n = 0, for n > 0
00 = undefined/indeterminate
Also:
ax = ay → x = y
requires appropriate base conditions, typically:
a > 0 and a ≠ 1
If a = 1, the exponents tell you nothing.
COMPARING POWERS / NUMBERS
- NUMBERS BETWEEN 0 AND 1 ⭐
0 < x < 1
x2 < x < √x < 1 < 1/x
So:
• Squaring makes x smaller.
• Square root makes x larger but still < 1.
• Reciprocal jumps above 1.
- NUMBERS GREATER THAN 1
x > 1
1/x < 1 < √x < x < x2
So:
• Squaring makes x larger.
• 2x > x.
• Reciprocal jumps below 1.
Key saved idea:
Whether x > 1 or 0 < x < 1:
Taking the reciprocal jumps to the opposite side of 1.
Exception: x = 1.
- COMPARING NEGATIVE NUMBERS
Greatest negative number → remove negatives → choose SMALLEST magnitude.
Least negative number → remove negatives → choose LARGEST magnitude.
Example:
-2, -5, -9
Greatest = -2
Least = -9
- COMPARING NUMBERS WITH NEGATIVE POWERS
a−n = 1/an
Then compare.
- FRACTIONAL EXPONENTS — COMMON POWER
- ORDER PRESERVATION UNDER SQUARING ⭐
If:
0 < a < b
Then:
a2 < b2
Order preserved.
Both negative:
If:
a < b < 0
Then:
a2 > b2
Order reverses.
Mixed signs:
No fixed comparison rule from squaring alone.
Recognition: Check signs before squaring to compare.
FRACTION / ROOT ALGEBRA COMPARISON
- ROOT + FRACTION COMPARISON ⭐
√a / b = √(a/b2)
a / √b = √(a2/b)
Then compare the radicands.
Since √x is increasing for x ≥ 0:
Larger radicand → larger square root.
FRACTION MOVEMENT
- SAME NUMBER ADDED TO NUMERATOR & DENOMINATOR ⭐
a/b
Adding k > 0:
(a + k)/(b + k)
moves the fraction toward 1.
If:
a/b < 1
Then:
(a + k)/(b + k) > a/b
If:
a/b > 1
Then:
(a + k)/(b + k) < a/b
Memory:
Adding the same positive amount to the top and bottom pulls the fraction toward 1.
- COMPARING POSITIVE FRACTIONS — CROSS MULTIPLY
ad > bc → a/b > c/d
ad < bc → a/b < c/d
- NUMERATOR / DENOMINATOR SIZE RECOGNITION
Increasing the numerator → increases the fraction.
Increasing the denominator → decreases the fraction.
So, all else equal:
Larger numerator + smaller denominator → larger fraction.
This applies whether the fractions are below or above 1.
CUSTOM / INTEGER FUNCTION
- GREATEST INTEGER / FLOOR FUNCTION ⭐
Examples:
⌊3.8⌋ = 3
⌊-2.3⌋ = -3
Important: For negative numbers, ⌊-2.3⌋ = -3, NOT -2.
MASTER ALGEBRA RECOGNITION
When you see:
• Fixed sum + maximum product → equalize.
• Fixed product + minimum sum → equalize.
• x + k/x, with x > 0 → AM-GM.
• Largest k with Expression ≥ k → find the expression's minimum.
• |x - a||x + a| → |x2 - a2|.
• Sum of many absolute values → median.
• Square root equation → remember the root is nonnegative; verify after squaring.
• Same exponent → combine bases.
• Repeated ax → substitute.
• Given ax, asked for related base → rewrite the base; don't solve x.
• Base > 1 → exponent inequality direction stays the same.
• 0 < base < 1 → exponent inequality direction reverses.
• 0 < x < 1 → x2 < x < √x < 1 < 1/x.
• x > 1 → 1/x < 1 < √x < x < x2.
• Compare by squaring → check signs first.
• Roots + fractions → put everything under one square root.
• Same amount added to numerator and denominator → fraction moves toward 1.
• Floor of a negative decimal → move DOWN to the next smaller integer.
Moderator:
