Some Important Formulas for Quadratic Equation and exponential

(i) $\displaystyle a^{m} . a^{n} = a^{m+n}$

(ii) $\displaystyle \frac{ a^{m}}{a^{n}} = a^{m- n}$

(iii) $\displaystyle (a^{m})^{n} = a^{mn}$

(iv) $\displaystyle a^{-n} = \frac{1}{a^{n}}$

(v) $\displaystyle \sqrt[n]{a^{m}}= \displaystyle  a^{\frac{m}{n}}$

(vii)$\displaystyle  (ab)^{m} = a^{m}. b^{m}$.

(viii)$\displaystyle \left (\frac{a}{b} \right )^{m} = \frac{a^{m}}{b^{m}}$

(ix) If $\displaystyle a^{m} = b^{m} (m ≠ 0), \;\;then\;\; a = b.$

(x) If $\displaystyle  a^{m}= a^{n}\;\; then\;\; m = n.$

● Quadratic Equation Formulas :

Quadratic equation is of form  $ax² + bx + c = 0 .$....*

(i) Roots of the quadratic equation are $\alpha, \beta = \displaystyle \frac{-b ± \sqrt{(b^{2} – 4ac)}}{2a}$.

(ii) If $\alpha$ and $\beta$ be the roots of the quadratic equation  then,

sum of its roots  $= \alpha+\beta = \displaystyle \frac{-b}{a}$  or we can say $\displaystyle \frac{- (\text{coefficient of x})}{\text{(coefficient of \(x^2\) )}}$

and product of its roots  $= \alpha.\beta = \displaystyle \frac{c}{a}$ or we can say   $\displaystyle \frac{\text{Constant term}} {\text{(Coefficient of \(x^{2}\))}}$.

(iii) The quadratic equation with both real  roots are $\alpha$ and $\beta$ is

$x^2 - (\alpha + \beta)x + \alpha.\beta= 0$

i.e. , $x^2 - (\text{sum of the roots}) x + \text{product of the roots} = 0.$

(iv) The expression $(b^2 - 4ac)$ is called the discriminant of quadratic equation. Denoted by $D$

(v) If $a, b, c$ are real and rational then the roots of quadratic equation are

(a) Real and equal when $b^2 - 4ac = 0$.

(b) Real and distinct when $b^2 - 4ac > 0$.

(c) Imaginary when $b^2 - 4ac < 0$.

(d)Rational when $b^2- 4ac$ is a perfect square and

(e) Irrational when $b^2 - 4ac$ is not a perfect square.

(vi) If $\alpha + i\beta$ be one root of quadratic equation then its other root will be complex conjugate  of this i.e $\alpha - i\beta$ .

(vii) If $\alpha +\sqrt{\beta}$ be one root of equation quadratic equation then its other root will be conjugate irrational quantity $\alpha - \sqrt{\beta}$ (a, b, c are rational).

● Arithmetical Progression (A.P.):

(i) The general form of an A. P. is $a, a + d, a + 2d, a + 3d,.....$

where a is the first term and d, the common difference of the A.P.

(ii) The nth term of the above A.P. is $a_{n} = a + (n - 1)d.$

(iii) The sum of first n terns of the above A.P. is $\displaystyle S_{n} = \frac{n}{2} (a + l)$  (where l is last term )or, $\displaystyle S_{n} = \frac{n}{2} [2a + (n - 1) d]$

(iv) The arithmetic mean between two given numbers a and b is $\displaystyle \frac{(a + b)}{2}$.

(v) $1 + 2 + 3 + ...... + n =\displaystyle \frac{ n(n + 1)}{2}$.

(vi) $1^{2} + 2^{2} + 3^{2} +… + n^{2} = \displaystyle\frac{n(n+ 1)(2n+ 1)}{6}.$

(vii) $1^{3} + 2^{3} + 3^{3} + . . . . + n^{3}= \displaystyle \frac{{n^{2}(n + 1)^{2}}}{4 }$.

● Geometrical Progression (G.P.) :

(i) The terms G.P. is $a, ar, ar^{2}, ar^{3}, . . . . .$  where a is the first term and r, the common ratio of the G.P.

(ii) The nth term given  G.P. is $t_{r} = a.r^{n-1}$ .

(iii) The sum of first n terms of the above G.P. is $S_{n} =\displaystyle  \frac{a . (1 - r^{n})}{(1 – r)}$when $-1 < r < 1$

or, $S _{n}= \displaystyle \frac{a .(r^{n} – 1)}{(r – 1) }$ when $r > 1$ or $r < -1$

(iv) $a + ar + ar^{2} +........+\infty = \displaystyle \frac{a}{1 – r}$ where $-1 < r < 1$.