Quantitative Aptitude ALGEBRA for IBPS PO & CLERK
- (a + b)2 = a2 + b2 + 2ab
- (a – b)2 = a2 + b2 – 2ab
- (a + b)3 = a3 + b3 + 3ab (a + b)
- (a – b)3 = a3 – b3 – 3ab (a – b)
- a2 – b2 = (a – b) (a + b)
- (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
- a3 + b3 = (a + b) (a2 – ab + b2)
- a3 – b3 = (a – b) (a2 + ab + b2)
- a3 + b3 + c3 – 3abc = (a + b + c) (a2 + b2 + c2 – ab – bc – ca)
if a + b + c = 0
- a3 + b3 + c3 = 3abc
Remainder theorem: If f(x) is divided by(x – a), then the remainder is f(a).
Factor theorem: If (x – a) is the factor of f(x), then f(a) = 0.
A Linear equation is one in which each variable occurs only in its first power & not in any higher powers. Generally, we have
- One equation in one unknown
- Two equations in two unknowns
- Three equations in three unknown
The general form of a system of linear equations in two variables x and y is
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
Set of Equations:
- Consistent set of equations: Those set of equations which have solutions.
Further, these set of equations can have a unique solution if
Condition: a1 / a2 ≠ b1 / b2
These set of equations can have infinite number of solutions if
Condition: a1 / a2= b1 / b2= c1 / c2
- Inconsistent set of equations: Those set of equations which have no solution.
Condition: a1 / a2 = b1 / b2 ≠ c1 / c2
A Quadratic equation is one in which each variable occurs only in its second power.
General form of quadratic equation is
ax2 + bx + c = 0
Roots are– b + / 2a, – b – / 2a
Sum of roots = -b / a
Product of roots = c /a
Nature of roots
- If b2 – 4ac = 0 roots are real and equal
- b2 – 4ac > 0 roots are real and distinct
- b2 – 4ac < 0 roots are imaginary
Forming Equation from roots:
If α and β are the roots of any quadratic equation then that equation can be written in the form
x2 – (α + β)x + αβ = 0
i.e. x2 – (sum of the roots) x + Product of the roots = 0.
Maximum /Minimum value of Quadratic Equation:
The equation ax2 + bx + c = 0, will have maximum value when a < 0 and minimum value when a > 0.
The maximum or minimum values are given by
4ac − b2 / 4a
, and will occur at x =-b / 2a
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