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.
Step-by-step explanation: formulas –
a2 – b2 = (a – b)(a + b)(a + b)2 = a2 + 2ab + b2a2 + 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 – 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 + b4a4 – 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 = amnFractional 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 discriminantFor real and distinct roots, Δ > 0For real and coincident roots, Δ = 0For non-real roots, Δ < 0If 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).nn! = 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\
n! = (1).(2).(3)…..(n − 1).nn! = n(n − 1)! = n(n − 1)(n − 2)! = ….0! = 1\((a + b)^{n} = a^{n}+na^{n-1}b+\frac{n(n-1)}{2!}a^{n-2}b^{2}+\frac{n(n-1)(n-2)}{3!}a^{n-3}b^{3}+….+b^{n}, where\;,n>1\)