### Algebra Formulas – What is Algebra?

Algebra is basically a branch of mathematics which deals with symbols and the rules for manipulating with those symbols. Algebra allows you to substitute the values in order to solve the equations for the unknown quantities.

### Algebra Formula

Algebra Formulas represents the relationship between the different variables. The variable can be taken as a, b, c, x, y or any other alphabet that represents a number unknown yet.

#### A list of Algebraic formulas

• a2 – b2 = (a – b)(a + b)
• (a+b)2 = a2 + 2ab + b2
• a2 + b2 = (a + b)2 – 2ab
• (a – b)2 = a2 – 2ab + b2
• (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
• (a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca
• (a + b)3 = a3 + 3a2b + 3ab2 + b3 ; (a + b)3 = a3 + b3 + 3ab(a + b)
• (a – b)3 = a3 – 3a2b + 3ab2 – b3
• a3 – b3 = (a – b)(a2 + ab + b2)
• a3 + b3 = (a + b)(a2 – ab + b2)
• (a + b)3 = a3 + 3a2b + 3ab2 + b3
• (a – b)3 = a3 – 3a2b + 3ab2 – b3
• (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4)
• (a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4)
• a4 – b4 = (a – b)(a + b)(a2 + b2)
• a5 – b5 = (a – b)(a4 + a3b + a2b2 + ab3 + b4)
• If n is a natural number an – bn = (a – b)(an-1 + an-2b+…+ bn-2a + bn-1)
• If n is even (n = 2k), an + bn = (a + b)(an-1 – an-2b +…+ bn-2a – bn-1)
• If n is odd (n = 2k + 1), an + bn = (a + b)(an-1 – an-2b +…- bn-2a + bn-1)
• (a + b + c + …)2 = a2 + b2 + c2 + … + 2(ab + ac + bc + ….)
• Laws of Exponents (am)(an) = am+n ; (ab)m = ambm ; (am)n = amn
• Fractional Exponents a = 1 ; aman=amn ; am = 1am ; am = 1am

• For a quadratic equation ax2 + bx + c where a ≠ 0, the roots will be given by the equation as $\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$
• Δ = b2 − 4ac is called the discrimination
• For real and distinct roots, Δ > 0
• For real & coincident roots, Δ = 0
• For non-real roots, Δ < 0
• If α and β are the two roots of the equation ax2 + bx + c then, α + β = (-b / a) and α × β = (c / a).
• If the roots of a quadratic equation are α and β, the equation will be (x − α)(x − β) = 0

#### Factorials

• n! = (1).(2).(3)…..(n − 1).n
• n! = n(n − 1)! = n(n − 1)(n − 2)! = ….
• 0! = 1
• $\ (a + b)^{n} = a^{n}+na^{n-1}b+\frac{n(n-1)}{2!}a^{n-2}b^{2}+\frac{n(n-1)(n-2)}{3!}a^{n-3}b^{3}+….+b^{n}, where\;,n>1$

### Algebra Problems with a Solution

#### Solved Examples of Algebra Formulas

Question 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

Question 2:
43 × 42 = ?
Solution:
Using the exponential formula (am)(an) = am+n
where a = 4
43 × 42
= 43+2 = 45 = 1024