Page 36 - CHAPTER 4 (Quadratic equations)

CHAPTER 4

QUADRATIC EQUATIONS

• A quadratic equation whose roots are the reciprocal of the roots of

1
2
equation ax +bx+c=0 (i.e., the roots are 1 , ) is given by
α β
1
1
2
a( ) +b( )+c=0.

• A quadratic equation whose roots are p times the roots of the
equation ax +bx+c=0 (i.e., pα,pβ) is given by a(x/p) +b(x/p)+c=0.
2
2
• A quadratic equation whose roots are square of the roots of the

2
2 2
2
2
equation ax +bx+c=0 (i.e. roots are α ,β ) is given by a x +(2ac-
2 2
b )x+c =0 .
D. Maximum and Minimum Value of a Quadratic Expression ax2 +

bx + c

• At x=-b/2a, the maximum or minimum value of the quadratic

expression is attained.

• When a>0 the expression attains minimum value ,y= 4ac-b 2 .
4a

• When a<0 the expression attains maximum value ,y= 4ac-b 2 .
4a

E. Sign of Quadratic Expression f(x)=ax +bx+c:
2
• If , are the roots of the corresponding quadratic equation, then

for x=α and x=β, the value of the expression is equal to zero.

• But for every other x, the expression is either less than zero or

greater than zero i.e., f(x)<0 or f(x)>0.

• Thus the sign of ax +bx+c, x∈ , is determined by the following
2
rules:

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