Question

Write whether the following statemensst are true or false .Justify your answers :

1.Every quadratic equation has at least one real root

2.If in a quadratic equation of ,the coeffecient of x is zero then the quadratic equation has no real roots .

3.If the coeffecient of x2 and the constant term of the quadratic equation have opposite signs ,then the quadratic equation has real roots .

4.A quadratic equation with integeral coeffecient has integral roots.

Answer 1

1. False.
If the value of b2-4ac is negetive for a quadratic equation, it will have no real roots.

2. False.
Consider the equation, x2+0x-1 here the value of b2- 4ac=4 which is positive. So the quadratic equation has real roots.

3. True.
If the coefficient of x2 (a) and constant term (c) has opposite signs, then the value of ac will be negative. So -4ac will be positive. Thus the value of b2-4ac will be positive and thus the quadratic equation has only real roots.
4.False..
Consider the givn Quadratic equation...
8x2 - 2x -1 = 0
by splitting the middle term,
8x2 -4x + 2x -1 = 0
4x(2x-1)+1(2x-1)=0
(4x+1)(2x-1)=0
if 4x+1 =0
THEN , x = -1/4

if 2x-1 =0
then, x = 1/2
therefore, x = -1/4 , 1/2

so the givn equation has integral coefficients but no integral roots.
hence the given statemnt is false.