Algebra Formulas

Algebra is a branch of mathematics that substitutes letters for numbers. 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. The numbers are constants. Algebra also includes real numbers, complex numbers, matrices, vectors and much more. X, Y, A, B are the most commonly used letters that represent algebraic problems and equations.

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
  • (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 – b= a3 – b3 – 3ab(a – b)
  • a3 – b3 = (a – b)(a2 + ab + b2)
  • a3 + b3 = (a + b)(a2 – ab + b2)
  • (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 +an-3b2…- 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 a0 = 1 ; $\frac{a^{m}}{a^{n}} = a^{m-n}$ ; $a^{m}$ = $\frac{1}{a^{-m}}$ ; $a^{-m}$ = $\frac{1}{a^{m}}$
  • Roots of Quadratic Equation
    • For a quadratic equation ax2 + bx + c = 0 where a ≠ 0, the roots will be given by the equation as \(x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\)
    • Δ = b2 − 4ac is called the discriminant
    • For real and distinct roots, Δ > 0
    • For real and coincident roots, Δ = 0
    • For non-real roots, Δ < 0
    • If α and β are the two roots of the equation ax2 + bx + c = 0 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
    • \((a + b)^{n} = a^{n}+na^{n-1}b+\frac{n(n-1)}{2!}a^{n-2}b^{2}+\frac{n(n-1)(n-2)}{3!}a^{n-3}b^{3}+….+b^{n}, where\;,n>1\)

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 $\times$ 8
= 16

Example 2:
43 $\times$ 42 = ?
Solution:
Using the exponential formula (am)(an) = am+n
where a = 4
43 $\times$ 42
= 43+2
= 45
= 1024

29 Comments

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