Exercise - 2.4

. Construct a quadratic equation with roots and − . . A quadratic polynomial has one of its zeros + and it satisfies p ( ) = .

Find the quadratic polynomial. . If α and β are the roots of the quadratic equation x + x + = , form a quadratic polynomial with zeroes α , β . .

If one root of k ( x − ) = x − is double the other root, show that k = or − . . If the difference of the roots of the equation x − ( a + ) x + a − = is equal to their product, then prove that a = . .

Find the condition that one of the roots of ax + bx + c may be (i) negative of the other, (ii) thrice the other, (iii) reciprocal of the other. . If the equations x − ax + b = and x − ex + f = have one root in common and if the second equation has equal roots, then prove that ae = ( b + f ) . .

Discuss the nature of roots of (i) − x + x + = , (ii) x − x − = , (iii) x + x = . . Without sketching the graphs, find whether the graphs of the following functions will intersect the x -axis and if so in how many points. (i) y = x + x + , (ii) y = x − x − , (iii) y = x + x + .

. Write f ( x ) = x + x + in completed square form. . .

Quadratic Inequalities Here we shall learn to solve the quadratic inequalities ax + bx + c < or ax + bx + c > .