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5b2. Quadratic Equations and Inequalities

Degree of an equation: The highest power of the variable is called the degree of the equation.


For example,


1. x + 2 = 3 Þ equation of degree one.


2. x2 + 2x + 3 = 0 Þ equation of degree two.


3. x3 + x2 + 4x - 2 = 0 Þ equation of degree three.


4. xn + 2xn-2 + .... = 0 Þ equation of degree n.


An equation of degree two is called a Quadratic Equation.


A quadratic polynomial is any polynomial equivalent to one of the form ax2 + bx + c.


A quadratic equation is any equation equivalent to one of the form ax2 + bx + c = 0.


Roots of an equation: Roots are those values of the variable parameter, which satisfies the equation. Degree of an equation gives the number of maximum roots possible for that equation.


By using formula:


If there is a quadratic equation of the form, then the roots of the equation is given by,



Inequalities


As in equalities we get the exact value of the variables in inequalities we don't. In inequalities we can only get a range of values for the variable, which can satisfy the in equation. In inequalities we do a relation comparison between the variables.


For example, if a > b and b > c then a > c. Here we don't know the values of a, b and c but we can compare which one is greater than the other.


Examples

(A) 4 < x < 16
(B) –4 > x > –16
(C) 4 > x > –16
(D) x < 16
(E) –16 < x < 4



Þ -6 < (10 - x) < 6


Þ-16 < -x < -4    Þ 4 < x < 16    Hence, (A)









 
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