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QUADRATIC EQUATIONS

What are quadratic equations?

Any equation of the form

ax² + bx + c = 0

is called a quadratic equation or expression. Here a <> 0 and a, b, c are the elements of the set of complex numbers.

The values of x, which satisfy a quadratic expression, are called its roots. A quadratic expression always has two roots. The roots of any quadratic expression

ax² + bx + c = 0

has the solutions

Derivation

Consider the quadratic equation

ax² + bx + c = 0 

with . dividing both sides of the equation by a

on rearranging

completing the square by adding to both sides

taking square of both sides

or

this is the result : the Quadratic Formula , written normally as

Remark. The plus-minus sign states that you have two numbers

  and .
The value b² - 4ac is called the discriminant of the equation and is represented by D

these roots are referred to as a and b for the positive and negative values of D respectively.

The equation whose roots are a and b respectively is given by

 

(x- a) (y - b) = 0

Consider the quadratic equation

 ax² + bx + c = 0

A real number x will be called a solution or a root if it satisfies the equation,  ax² + bx + c = 0 . It is easy to see that the roots are exactly the x-intercepts of the quadratic function  f(x) = ax² + bx + c = 0  , that is the intersection between the graph of the quadratic function with the x-axis.

a<0	a>0

RELATION BETWEEN ROOTS AND COEFFICIENTS

1. the sum of the above roots is equal to the negative of coefficient of x divided by the coefficient of x², i.e.,

 

a + b = -b/a

2. the product of roots is equal to the constant term of the equation divided by the coefficient of x², i.e.,

 

a b = c/a

NATURE OF ROOTS

1. if D is >= 0, then the roots are real. If D > 0 then the roots are real and unequal. Also if D < 0 then the roots are real and equal.

2. If D< 0 then the roots are imaginary and unequal.

3. If D is a perfect square then the roots are rational, whereas if D is not a perfect square then the roots are irrational.

SIGN OF A QUADRATIC EXPRESSION

1. If a quadratic expression (x- a) (y - b), is positive then it means that either, x > b or x < a

This is because the above expression can be positive only when either both the factors are positive or both negative.

Thus we can say that the roots of the expression do not lie between a and b.

2. If a quadratic expression (x- a) (y - b), is negative then it means that either, x > a or x < b

This is because the above expression can be negative only when one of the factor is positive or negative.

Thus we can say that the roots of the expression lie in the interval (a, b).

NOTE: The expression has the same sign as that of a (for all real values of x), except when the roots of the equation are real and unequal, and x has a value lying between them.