Algebra turns words into equations — x, y stand for unknowns, and each equation
is a constraint. Linear equations are straight lines, quadratics are parabolas, inequalities
are half-planes. Learn to isolate, eliminate, and read roots without solving fully.
Highest power is 1 — a straight line. Standard form:
ax + b = 0.
How to solve (isolate x):
3(x−2) → 3x−6.x-terms on one side, numbers
on the other.
x alone.2x + 3 = 11 → subtract 3 → 2x=8 → divide by
2 → x=4. Check: 2×4+3=11 ✓. Every step preserves equality — what you
do to one side, do to the other.
Two unknowns need two equations for a unique solution. Two standard methods:
x = y + 1).
x + y = 7, 2x − y = 5 — add directly (y
cancels): 3x=12 → x=4, then y=3. Elimination is faster when
coefficients are already opposites; substitution is safer when one variable is already
isolated.0 = non-zero or 0=0.
Highest power is 2 — a parabola. Standard form:
ax² + bx + c = 0, a ≠ 0.
Find two numbers that multiply to c (or a×c when a≠1)
and add to b.
x² −5x +6=0 — numbers that multiply to 6 and add to −5
are −2 and −3 → (x−2)(x−3)=0 → x=2, 3.2x²+7x+3=0: need a×c=6 and b=7 → 6 and 1 → split
middle: 2x²+6x+1x+3=2x(x+3)+1(x+3)=(2x+1)(x+3) → x=−1/2, −3.
x = [ −b ± √(b² − 4ac) ] / 2a
Discriminant Δ = b² − 4ac determines nature without solving:
| Δ | Roots |
|---|---|
Δ > 0 |
Two distinct real roots |
Δ = 0 |
One repeated real root (−b/2a) — parabola touches x-axis |
Δ < 0 |
No real roots (two complex) — parabola floats above/below |
x²+2x+1=0 → Δ=4−4=0 → repeated root x=−1.
x²+x+1=0 → Δ=1−4=−3 <0 → no real root.
For ax²+bx+c=0 with roots α, β:
α + β = −b/aαβ = c/a
Useful to find roots without full formula: if you know sum and product, the quadratic is
x² − (sum)x + product = 0. Example: sum 5, product 6 → x²−5x+6=0.
xa × xb = xa+b
xa / xb = xa−b
(xa)b = xab
logb(x) = y ⇔ by = x
log(xy) = log x + log ylog(xn) = n·log xlog₂(x)=5 → 2⁵=x → 32.
See Module 20 for full log laws, change of base, and extraneous solutions.
Solve like equations, with one twist:
multiplying/dividing by a negative flips the sign
(> ↔ <, ≥ ↔ ≤).
−2x < 6 → divide by −2 (flip):
x > −3.2x + 3 ≥ 11 → 2x ≥ 8 → x ≥ 4 — no flip because divisor positive.
A function f(x) is a rule input→output.
f(3) means replace every x with 3. If
f(x)=2x+1, then f(3)=7.
f⁻¹(x): the rule that reverses f. Steps:
replace f(x) with y, swap x and y, solve
for new y.
f(x)=2x+3 → y=2x+3 → swap:
x=2y+3 → y=(x−3)/2 → f⁻¹(x)=(x−3)/2. Check:
f(f⁻¹(7))=f(2)=7 ✓.
AP: Tn=a+(n−1)d,
Sn=(n/2)[2a+(n−1)d]
GP: Tn=a·rn−1, infinite sum a/(1−r) when |r|<1. Full derivations,
telescoping, and HM are in Module 19.
| Symbol | Meaning |
|---|---|
x, y, z |
Variables (unknowns) |
a, b, c |
Coefficients |
f(x) |
Function |
d |
Common difference (AP) |
r |
Common ratio (GP) |
Δ |
Discriminant b²−4ac |
Standard linear form?
2x+3=11 → x?
x²−5x+6=0 → roots?
For ax²+bx+c=0, sum of roots = ?
Product of roots = ?
Δ = b²−4ac < 0 → roots?
log₂(x)=5 → x?
Solve −2x < 6 → ?
ax+b=0 ; isolate: simplify → move → divide
System: substitution (solve one, plug) or elimination (equalize coeff, add/sub)
ax²+bx+c=0 ; factoring: find pair multiply to a×c, add to b
Formula ⇒ (−b±√Δ)/2a ; Δ=b²−4ac
Δ>0:2 real ; Δ=0:1 repeated (−b/2a) ; Δ<0:no real
Vieta ⇒ sum=−b/a ; product=c/a
xa×xb=xa+b ; /→xa−b ; (xa)b=xab
logb(x)=y ⇔ by=x ; log(xy)=logx+logy ; log(xn)=n logx
Like equations but ×/÷ by negative → flip sign
f(a) ⇒ replace x with a ; inverse: y=f(x) → swap x,y → solve for y
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