1) Construct a quadratic equation with roots 7 and −3.
2) A quadratic polynomial has one of its zeros 1 +√5 and it satisfies p(1) = 2. Find the quadratic polynomial.
3) If α and β are the roots of the quadratic equation x^{2} + √2x+3 = 0, form a quadratic polynomial with zeroes 1/α, 1/β .
4) If one root of k(x − 1)^{2} = 5x − 7 is double the other root, show that k = 2 or −25.
5) If the difference of the roots of the equation 2x^{2} − (a+1)x + a −1 = 0 is equal to their product, then prove that a = 2.
6) Find the condition that one of the roots of ax^{2} + bx + c may be
(i) negative of the other,
(ii) thrice the other,
(iii) reciprocal of the other. Solution
7) If the equations x^{2} −ax+b = 0 and x^{2} −ex+f = 0 have one root in common and if the second equation has equal roots, then prove that ae = 2(b + f). Solution
8) Discuss the nature of roots of
(i) −x^{2} + 3x+1 = 0,
(ii) 4x^{2} − x − 2 = 0,
(iii) 9x^{2} + 5x = 0. Solution
9) 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^{2} + x + 2
(ii) y = x^{2} − 3x − 7
(iii) y = x^{2} + 6x + 9. Solution
10) Write f(x) = x^{2} + 5x + 4 in completed square form.
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