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Algebra formulas

50+

Total 50+ Formulas
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An algebraic equation depicts a
scale, what is done on one side of the scale with a number is also done to either side of the scale
...
Algebra also includes real numbers, complex numbers, matrices, vectors and
much more
...

Algebra
Formulas from
Class 8 to Class
12

Algebra
Algebra
Algebra
Algebra
Algebra
Formulas For Formulas For Formulas For Formulas For Formulas For
Class 8
Class 9
Class 10
Class 11
Class 12

Important Formulas in Algebra
Here is a list of Algebraic formulas –


a2 – b2 = (a – b)(a + b)



(a + b)2 = a2 + 2ab + b2



a2 + b2 = (a + b)2 – 2ab

Reference: Books,
"Elementary Algebra" by Charles P
...
khanacademy
...
math
...
edu)

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)



Laws of Exponents (am)(an) = am+n ; (ab)m = ambm ; (am)n = amn



Fractional Exponents a0 = 1 ;

aman=am−n;am=1a−m;a−m=1am


Roots of Quadratic Equation


For a quadratic equation ax2 + bx + c = 0 where a ≠ 0, the roots will be given by the
equation as

x=−b±b2−4ac2a
Reference: Books,
"Elementary Algebra" by Charles P
...
khanacademy
...
math
...
edu)

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

If the roots of a quadratic equation are α and β, the equation will be (x − α)(x − β) = 0

Factorials


n! = (1)
...
(3)…
...
n



n! = n(n − 1)! = n(n − 1)(n − 2)! = …
...
+bn,where,n>1

Read more:


Algebra



Factorial



Maths models



Maths worksheets

Solved Examples
Example 1: Find out the value of 52 – 32
Solution:
Using the formula a2 – b2 = (a – b)(a + b)
where a = 5 and b = 3
(a – b)(a + b)
= (5 – 3)(5 + 3)
= 2×8= 16
Reference: Books,
"Elementary Algebra" by Charles P
...
khanacademy
...
math
...
edu)

Algebra formulas

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Basic Algebraic Identities
These are the foundational formulas for simplifying expressions and solving equations:
1
...

3
...


(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2(a+b)2=a2+2ab+b2
(a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2(a−b)2=a2−2ab+b2
(a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2(a+b)(a−b)=a2−b2
(x+a)(x+b)=x2+(a+b)x+ab(x + a)(x + b) = x^2 + (a+b)x + ab(x+a)(x+b)=x2+(a+b)x+ab

2
...

2
...

4
...
Factoring Formulas
Common formulas for factoring expressions:
1
...
x2+(a+b)x+ab=(x+a)(x+b)x^2 + (a + b)x + ab = (x + a)(x + b)x2+(a+b)x+ab=(x+a)(x+b)

Reference: Books,
"Elementary Algebra" by Charles P
...
khanacademy
...
math
...
edu)

Algebra formulas

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a3+b3+c3−3abc=(a+b+c)(a2+b2+c2−ab−bc−ca)a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2
+ b^2 + c^2 - ab - bc - ca)a3+b3+c3−3abc=(a+b+c)(a2+b2+c2−ab−bc−ca) (if a+b+c=0a
+ b + c = 0a+b+c=0)

4
...
Arithmetic and Geometric Progression
Formulas for series and sequences:




Arithmetic Progression (AP):
1
...
Sum of nnn terms: Sn=n2[2a+(n−1)d]S_n = \frac{n}{2}[2a + (n-1)d]Sn=2n
[2a+(n−1)d]
Geometric Progression (GP):
1
...
Sum of nnn terms (finite): Sn=a1−rn1−rS_n = a \frac{1 - r^n}{1 - r}Sn=a1−r1−rn
(if r≠1r \neq 1r =1)
3
...
Exponent and Logarithm Rules
These rules simplify operations with exponents and logarithms:




Exponent Rules:
1
...
aman=am−n\frac{a^m}{a^n} = a^{m-n}anam=am−n
3
...
a0=1a^0 = 1a0=1
5
...
log⁡(ab)=log⁡a+log⁡b\log(ab) = \log a + \log blog(ab)=loga+logb

Reference: Books,
"Elementary Algebra" by Charles P
...
khanacademy
...
math
...
edu)

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log⁡(ab)=log⁡a−log⁡b\log\left(\frac{a}{b}\right) = \log a - \log blog(ba
)=loga−logb
3
...
log⁡ab=log⁡blog⁡a\log_a b = \frac{\log b}{\log a}logab=logalogb

7
...
Slope (mmm):
m=−abm = -\frac{a}{b}m=−ba
2
...
Slope-Intercept Form:
y=mx+cy = mx + cy=mx+c

8
...
Inequalities
For real numbers a,b,ca, b, ca,b,c:
1
...

Reference: Books,
"Elementary Algebra" by Charles P
...
khanacademy
...
math
...
edu)

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Multiplying or dividing by a negative number reverses the inequality
...
The quadratic discriminant rule: For ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0,
o If b2−4ac>0b^2 - 4ac > 0b2−4ac>0: Two distinct real roots
...

o If b2−4ac<0b^2 - 4ac < 0b2−4ac<0: No real roots
...
Matrix and Determinants
1
...
Inverse of a 2x2 Matrix:
A−1=1det⁡(A)[d−b−ca],if det⁡(A)≠0A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d &
-b \\ -c & a \end{bmatrix}, \quad \text{if } \det(A) \neq 0A−1=det(A)1[d−c−ba
],if det(A) =0

11
...
(x+y+z)2=x2+y2+z2+2xy+2yz+2zx(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz +
2zx(x+y+z)2=x2+y2+z2+2xy+2yz+2zx
2
...
x3+y3+z3−3xyz=(x+y+z)(x2+y2+z2−xy−yz−zx)x^3 + y^3 + z^3 - 3xyz = (x + y +
z)(x^2 + y^2 + z^2 - xy - yz - zx)x3+y3+z3−3xyz=(x+y+z)(x2+y2+z2−xy−yz−zx)
o If x+y+z=0x + y + z = 0x+y+z=0, then x3+y3+z3=3xyzx^3 + y^3 + z^3 =
3xyzx3+y3+z3=3xyz
...
Partial Fractions
Used to break down a rational expression into simpler fractions:
For a rational function:

Reference: Books,
"Elementary Algebra" by Charles P
...
khanacademy
...
math
...
edu)

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\frac{P(x)}{Q(x)} = \frac{A}{(x
- a)} + \frac{B}{(x - b)} \quad \text{(if \(Q(x)\) is factored into linear terms)}
...


13
...
Sum of roots: −ba-\frac{b}{a}−ab
2
...
Sum of roots: −ba-\frac{b}{a}−ab
2
...
Product of roots: −da-\frac{d}{a}−ad

14
...
i2=−1i^2 = -1i2=−1 (where iii is the imaginary unit)
...
Conjugate of a complex number z=a+biz = a + biz=a+bi: z‾=a−bi\overline{z} = a biz=a−bi
...
Modulus of z=a+biz = a + biz=a+bi: ∣z∣=a2+b2|z| = \sqrt{a^2 + b^2}∣z∣=a2+b2
...
Multiplication of two complex numbers: (a+bi)(c+di)=(ac−bd)+(ad+bc)i
...
(a+bi)(c+di)=(ac−bd)+(ad+bc)i
...
Logarithmic Properties
Additional logarithmic properties for advanced problems:
1
...
\log_a b = \frac{\log b}{\log a}
...

2
...
a^{\log_a x} = x
...

3
...
\log_a 1 = 0 \quad \text{(for any base \(a >
0\))}
...


16
...
McKeague, Algebra for College Students" by Mark Dugopolski, "Higher Algebra" by Hall &
Knight, "College Algebra" by James Stewart, Lothar Redlin, and Saleem Watson, "Schaum's Outline of College Algebra" by
Murray Spiegel, Khan Academy (www
...
org), Paul's Online Math Notes (tutorial
...
lamar
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Page 9 of 21
For solving two linear equations:
1
...

2
...

3
...


17
...
For a 3x3 matrix:
det⁡(A)=a(ei−fh)−b(di−fg)+c(dh−eg),\det(A) = a(ei − fh) − b(di − fg) + c(dh −
eg),det(A)=a(ei−fh)−b(di−fg)+c(dh−eg),
where A=[abcdefghi]A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i
\end{bmatrix}A=adgbehcfi
...
Cofactor expansion:
det⁡(A)=∑j=1naijCij,\det(A) = \sum_{j=1}^n a_{ij}C_{ij},det(A)=j=1∑naijCij,
where CijC_{ij}Cij is the cofactor of aija_{ij}aij
...
Parabola, Ellipse, and Hyperbola (Conic Sections)
Equations of conic sections:
1
...

2
...

3
...


19
...
McKeague, Algebra for College Students" by Mark Dugopolski, "Higher Algebra" by Hall &
Knight, "College Algebra" by James Stewart, Lothar Redlin, and Saleem Watson, "Schaum's Outline of College Algebra" by
Murray Spiegel, Khan Academy (www
...
org), Paul's Online Math Notes (tutorial
...
lamar
...
Page 10 of 21
1
...
|x| = a \implies x = a \text{ or } x = -a
...

2
...
|a + b| \leq |a| + |b| \quad
\text{(Triangle Inequality)}
...

3
...


20
...
Polynomial division:
o For P(x)P(x)P(x) divided by (x−a)(x - a)(x−a), the remainder is P(a)P(a)P(a)
...
Factor theorem:
o If P(a)=0P(a) = 0P(a)=0, then (x−a)(x - a)(x−a) is a factor of P(x)P(x)P(x)
...
Arithmetic-Geometric Means
1
...
AM = \frac{a_1 + a_2 +
\dots + a_n}{n}
...

2
...
GM = \sqrt[n]{a_1 \cdot a_2
\cdot \dots \cdot a_n}
...

3
...


22
...
(a+b)mod  n=[(amod  n)+(bmod  n)]mod  n(a + b) \mod n = [(a \mod n) + (b \mod n)]
\mod n(a+b)modn=[(amodn)+(bmodn)]modn
...
(a⋅b)mod  n=[(amod  n)⋅(bmod  n)]mod  n(a \cdot b) \mod n = [(a \mod n) \cdot (b \mod
n)] \mod n(a⋅b)modn=[(amodn)⋅(bmodn)]modn
...
Exponential Equations
1
...

2
...
a^x
= b \implies x = \log_a b
...


24
...
McKeague, Algebra for College Students" by Mark Dugopolski, "Higher Algebra" by Hall &
Knight, "College Algebra" by James Stewart, Lothar Redlin, and Saleem Watson, "Schaum's Outline of College Algebra" by
Murray Spiegel, Khan Academy (www
...
org), Paul's Online Math Notes (tutorial
...
lamar
...
Page 11 of 21
1
...
S = \frac{n}{6}[2a^2 + (n-1)d^2 +
2a(n-1)d]
...

2
...
S = \left(\frac{n}{2}[2a + (n1)d]\right)^2
...


25
...
Sum of roots: r1+r2+⋯+rn=−an−1an
...
r1
+r2+⋯+rn=−anan−1
...
Sum of product of roots taken two at a time: r1r2+r1r3+⋯=an−2an
...
r1r2+r1r3+⋯=anan−2
...
Product of all roots: r1r2…rn=(−1)na0an
...
r1r2
…rn=(−1)nana0
...
Partial Sum of Powers
1
...
S_n = \frac{n(n+1)}{2}
...

2
...
S_n =
\frac{n(n+1)(2n+1)}{6}
...

3
...
S_n =
\left(\frac{n(n+1)}{2}\right)^2
...


27
...
Sum of nnn-th powers of roots:
Sn=r1n+r2n+⋯+rkn(derived from Newton’s identities)
...
Sn=r1n+r2n+⋯+rkn
(derived from Newton’s identities)
...
Factorization of higher powers:
o a4+b4=(a2+b2)2−2a2b2
...
a4+b4=(a2+b2)2−2a2b2
...
a^5 - b^5 = (a - b)(a^4 + a^3b + a^2b^2 +
ab^3 + b^4)
...


28
...
McKeague, Algebra for College Students" by Mark Dugopolski, "Higher Algebra" by Hall &
Knight, "College Algebra" by James Stewart, Lothar Redlin, and Saleem Watson, "Schaum's Outline of College Algebra" by
Murray Spiegel, Khan Academy (www
...
org), Paul's Online Math Notes (tutorial
...
lamar
...
Page 12 of 21
Methods for solving:
1
...

2
...

3
...


29
...
Find the roots x1,x2x_1, x_2x1,x2
...
Test intervals (−∞,x1),(x1,x2),(x2,∞)(-\infty, x_1), (x_1, x_2), (x_2, \infty)(−∞,x1),(x1
,x2),(x2,∞) to determine where the inequality holds
...
Vector Algebra Basics
For vectors a⃗\vec{a}a and b⃗\vec{b}b:
1
...
\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}|
\cos\theta
...

2
...

3
...
|\vec{a}| = \sqrt{a_x^2 + a_y^2 +
a_z^2}
...


31
...
If two rows (or columns) of a determinant are identical, the determinant is 0
...
If a row (or column) is multiplied by a scalar kkk, the determinant is multiplied by kkk
...
Swapping two rows (or columns) changes the sign of the determinant
...
Cramer’s Rule
For a system of nnn linear equations AX=BAX = BAX=B:
Reference: Books,
"Elementary Algebra" by Charles P
...
khanacademy
...
math
...
edu)

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33
...
Matrix multiplication: (AB)ij=∑k=1nAikBkj
...
(AB)ij=k=1∑nAikBkj
...
Transpose of a matrix: (AT)ij=Aji
...
(AT)ij=Aji
...
Inverse of a matrix: A−1A=I(identity matrix)
...
A−1A=I(identity matrix)
...
Advanced Exponential Equations
1
...
x = \frac{\log b}{\log a}
...

2
...


35
...
General form of a geometric series: S=a+ar+ar2+⋯+arn−1=a(1−rn)1−r(r≠1)
...
S=a+ar+ar2+⋯+arn−1=1−ra(1−rn)(r =1)
...
Binomial series for (1+x)n(1 + x)^n(1+x)n:
(1+x)n=1+nx+n(n−1)2!x2+…(valid for ∣x∣<1)
...
(1+x)n=1+nx+2!n(n−1)x2+…(valid for ∣x∣<1)
...
Algebraic Methods for Optimization
1
...
x = -\frac{b}{2a}
...

2
...


37
...
McKeague, Algebra for College Students" by Mark Dugopolski, "Higher Algebra" by Hall &
Knight, "College Algebra" by James Stewart, Lothar Redlin, and Saleem Watson, "Schaum's Outline of College Algebra" by
Murray Spiegel, Khan Academy (www
...
org), Paul's Online Math Notes (tutorial
...
lamar
...
Page 14 of 21
For a polynomial P(x)=anxn+an−1xn−1+⋯+a0P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots +
a_0P(x)=anxn+an−1xn−1+⋯+a0:
Any rational root is of the form:
pq,\frac{p}{q},qp,
where ppp divides the constant term a0a_0a0 and qqq divides the leading coefficient ana_nan
...
Advanced Modular Arithmetic
1
...
a^{p-1} \equiv 1 \pmod{p} \quad
\text{(if \(p\) is prime)}
...

2
...
\phi(n) = \text{Number of
integers \(k\) such that \(1 \leq k \leq n\) and \(\gcd(k, n) =
1\)}
...


39
...
Inequalities involving absolute values: ∣x+y∣≤∣x∣+∣y∣(Triangle Inequality)
...
∣x+y∣≤∣x∣+∣y∣(Triangle Inequality)
...
Square root property: ∣x∣=x2
...
∣x∣=x2
...
Advanced Algebraic Concepts
1
...

2
...
x = f(t),
\quad y = g(t)
...


41
...
Expansion of (a+b)n(a + b)^n(a+b)n:
(a+b)n=∑k=0n(nk)an−kbk,(a + b)^n = \sum_{k=0}^n \binom{n}{k}a^{nk}b^k,(a+b)n=k=0∑n(kn)an−kbk,
Reference: Books,
"Elementary Algebra" by Charles P
...
khanacademy
...
math
...
edu)

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2
...
T_{\text{middle}} =
\binom{n}{\frac{n}{2}}a^{\frac{n}{2}}b^{\frac{n}{2}}
...

3
...
(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \dots
\quad (|x| < 1)
...


42
...
Sum of the ppp-th power: Sp(n)=1p+2p+⋯+np,S_p(n) = 1^p + 2^p + \dots + n^p,Sp
(n)=1p+2p+⋯+np, where Sp(n)S_p(n)Sp(n) can be expressed recursively in terms of
lower powers
...
Roots of Unity
For the nnn-th roots of unity:
1
...

2
...


44
...
For a square matrix AAA, eigenvalues λ\lambdaλ satisfy:
Reference: Books,
"Elementary Algebra" by Charles P
...
khanacademy
...
math
...
edu)

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...

2
...
A\vec{v} = \lambda \vec{v}
...


45
...
Rational functions:
f(x)=P(x)Q(x),f(x) = \frac{P(x)}{Q(x)},f(x)=Q(x)P(x),
where P(x)P(x)P(x) and Q(x)Q(x)Q(x) are polynomials, and Q(x)≠0Q(x) \neq 0Q(x) =0
...
Inverse functions: For y=f(x)y = f(x)y=f(x), the inverse is x=f−1(y)x = f^{1}(y)x=f−1(y), where:
f(f−1(x))=x
...
f(f−1(x))=x
...
Advanced Inequalities
1
...
\frac{x_1 + x_2 + \dots + x_n}{n} \geq \sqrt[n]{x_1x_2\dots
x_n}
...

2
...
\left(\sum_{i=1}^n a_i^2\right)\left(\sum_{i=1}^n
b_i^2\right) \geq \left(\sum_{i=1}^n a_ib_i\right)^2
...

3
...
f\left(\frac{x_1 + x_2 + \dots + x_n}{n}\right)
\leq \frac{f(x_1) + f(x_2) + \dots + f(x_n)}{n}
...

Reference: Books,
"Elementary Algebra" by Charles P
...
khanacademy
...
math
...
edu)

Algebra formulas

50+

Total 50+ Formulas
...
Properties of Modular Arithmetic
1
...
(p-1)! \equiv -1 \pmod{p}, \quad \text{if \(p\) is
a prime number
...

2
...


48
...
Nature of roots: For ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0,
o Discriminant Δ=b2−4ac\Delta = b^2 - 4acΔ=b2−4ac:
 Δ>0\Delta > 0Δ>0: Roots are real and distinct
...

 Δ<0\Delta < 0Δ<0: Roots are complex conjugates
...
Vertex form of a quadratic:
y=a(x−h)2+k,y = a(x - h)^2 + k,y=a(x−h)2+k,
where (h,k)(h, k)(h,k) is the vertex, and h=−b2ah = -\frac{b}{2a}h=−2ab, k=f(h)k =
f(h)k=f(h)
...
Logarithmic and Exponential Relationships
1
...

2
...
McKeague, Algebra for College Students" by Mark Dugopolski, "Higher Algebra" by Hall &
Knight, "College Algebra" by James Stewart, Lothar Redlin, and Saleem Watson, "Schaum's Outline of College Algebra" by
Murray Spiegel, Khan Academy (www
...
org), Paul's Online Math Notes (tutorial
...
lamar
...
Page 18 of 21
eln⁡x=xandln⁡(ex)=x
...
elnx=xandln(ex)=x
...
Symmetric Polynomials
A polynomial P(x1,x2,…,xn)P(x_1, x_2, \dots, x_n)P(x1,x2,…,xn) is symmetric if swapping any
two variables leaves PPP unchanged
...
Elementary symmetric polynomials:
o Sum of roots: e1=x1+x2+⋯+xne_1 = x_1 + x_2 + \dots + x_ne1=x1+x2+⋯+xn
...


51
...
Geometric Series Sum:
S=a1−r,if ∣r∣<1
...
S=1−ra,if ∣r∣<1
...
Harmonic Series:
Hn=1+12+13+⋯+1n
...
Hn
=1+21+31+⋯+n1
...
Alternating Series Test: If ∣an∣|a_n|∣an∣ decreases and lim⁡n→∞an=0\lim_{n \to \infty}
a_n = 0limn→∞an=0, then the series ∑(−1)nan\sum (-1)^n a_n∑(−1)nan converges
...
Advanced Progressions
1
...
a_n = a + (n-1)d
...

2
...
a_n = ar^{n-1}
...

3
...
McKeague, Algebra for College Students" by Mark Dugopolski, "Higher Algebra" by Hall &
Knight, "College Algebra" by James Stewart, Lothar Redlin, and Saleem Watson, "Schaum's Outline of College Algebra" by
Murray Spiegel, Khan Academy (www
...
org), Paul's Online Math Notes (tutorial
...
lamar
...
Page 19 of 21
an=1a+(n−1)d
...
an=a+(n−1)d1
...
Functional Equations
Functional equations involve functions f(x)f(x)f(x) satisfying conditions for all xxx:
1
...
f(x + y) = f(x) + f(y)
...

2
...
f(xy) = f(x)f(y)
...


54
...
Remainder Theorem:
P(a) is the remainder when P(x) is divided by (x−a)
...
P(a) is the remainder when P(x) is divided by (x−a)
...
Factor Theorem: (x−a)(x-a)(x−a) is a factor of P(x)P(x)P(x) if P(a)=0P(a) = 0P(a)=0
...
Probability in Algebra
1
...
P(E) =
\frac{\text{Number of favorable outcomes}}{\text{Total number of
outcomes}}
...

2
...
P(A \cup B) = P(A) + P(B) - P(A \cap
B)
...

Reference: Books:
Reference: Books,
"Elementary Algebra" by Charles P
...
khanacademy
...
math
...
edu)

Algebra formulas

50+

Total 50+ Formulas
...
"Elementary Algebra" by Charles P
...

2
...

3
...

4
...

5
...


Online Resources
1
...
khanacademy
...

2
...
math
...
edu)
A reliable resource for algebraic and calculus-based problem-solving
...
Wolfram MathWorld (mathworld
...
com)
Detailed mathematical explanations and derivations for various algebraic topics
...
Brilliant
...
brilliant
...

5
...
purplemath
...


Research Papers and Journals
1
...

2
...


Historical Reference
For classical algebraic techniques, Isaac Newton’s works on polynomials, symmetry, and series
are foundational
...
McKeague, Algebra for College Students" by Mark Dugopolski, "Higher Algebra" by Hall &
Knight, "College Algebra" by James Stewart, Lothar Redlin, and Saleem Watson, "Schaum's Outline of College Algebra" by
Murray Spiegel, Khan Academy (www
...
org), Paul's Online Math Notes (tutorial
...
lamar
...
Page 21 of 21


Scholarly articles analyzing Newton's work in mathematics and its relevance to algebra
...
Let me know!

Reference: Books,
"Elementary Algebra" by Charles P
...
khanacademy
...
math
...
edu)



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