# 43. Quadratic Equations Problems Part 3¶

1. If $$\alpha, \beta$$ are the roots of equation $$x^2 - px + q = 0,$$ show that $$\alpha + \frac{1}{\beta}$$ is a root of equation $$qx^2 - p(1 + q)x + (1 + q)^2 = 0$$
2. Determine the value of $$m$$ for which $$3x^2 + 4mx + 2 = 0$$ and $$2x^2 + 3x -2 = 0$$ may have a common root.
3. Find the value of $$a$$ if $$x^2 - 11x + a = 0$$ and $$x^2 - 14x + 2a = 0$$ have a common root.
4. If the equations $$ax^2 + bx + c = 0$$ and $$bx^2 + cx + a = 0$$ have a common root, then show that either $$a + b + c = 0$$ or $$a = b = c$$
5. Find the value of $$m$$ so that equations $$x^2 + 10x + 21 = 0$$ and $$x^2 + 9x + m = 0$$ may have a common root. Find also the equation formed by the other roots.
6. Show that the equations $$x^2 - x - 12 = 0$$ and $$3x^2 + 10x + 3 = 0$$ have a common root. Also find the common root.
7. If the equations $$3x^2 + px + 1 = 0$$ and $$2x^2 + qx + 1 = 0$$ have a common root, show that $$2p^2 + 3q^2 - 5pq + 1 = 0$$
8. Show that the equation $$ax^2 + bx + c = 0$$ and $$x^2 + x + 1 = 0$$ cannot have a common root unless $$a = b = c$$
9. If the equations $$x^2 + px + q = 0$$ and $$x^2 + p_1x + q_1 = 0$$ have a common root, show that it must be either $$\frac{pq_1 - p_1q}{q - q_1}$$ or $$\frac{q - q_1}{p_1 - p}$$
10. Prove that the two quadratic equations $$ax^2 + bx + c = 0$$ and $$2x^2 - 3x + 4 = 0$$ cannot have a common root unless $$6a = -4b = 3c$$
11. Prove that the equations $$(q - r)x^2 + (r - p)x + p - q = 0$$ and $$(r - p)x^2 + (p - q)x + q - r = 0$$ have a common root.
12. If the equations $$x^2 + abx + c = 0$$ and $$x^2 + acx + b = 0$$ have a common root, prove that their other roots satisfy the equation $$x^2 - a(b + c)x + a^2bc = 0$$
13. If the equations $$x^2 - px + q = 0$$ and $$x^2 - ax + b = 0$$ have a common root and the other root of the second equation is the reciprocal of the other root of the first, then prove that $$(q - b)^2 = bq(p - a)^2$$
14. Show that $$(x - 2)(x - 3) - 8(x - 1)(x - 3) + 9(x - 1)(x - 2) = 2x^2$$ is an identity.
15. Show that $$\frac{a^2(x - b)(x - c)}{(a - b)(a - c)} + \frac{b^2(x - a)(x - c)}{(b - a)(b - c)} + \frac{c^2(x - a)(x - b)}{(c - a)(c - b)} = x^2$$ is an identity.
16. Show that $$3x^{10} - 2x^5 + 8 = 0$$ is an equation.
17. Solve the equation $$\frac{x + 2}{x - 2} - \frac{x - 2}{x + 2} = \frac{5}{6}$$
18. Solve the equation $$\frac{2\sqrt{x} + 1}{3 - \sqrt{x}} = \frac{11 - 3\sqrt{x}}{5\sqrt{x} - 9}$$
19. Solve the equation $$(x + 1)(x + 2)(x - 3)(x - 4) = 336$$
20. Solve the equation $$\sqrt{x + 1} + \sqrt{2x - 5} = 3$$
21. Solve the equation $$2^{2x} + 2^{x + 2} - 32 = 0$$
22. A pilot flies an aircraft with a certain speed a distance of 800 km. He could have saved 40 minutes by increasing the average speed of the aircraft by 40 km/hour. Find the average speed of the aircraft.
23. The length of a rectangle is 2 meter more than its width. If the length is increased by 6 meter and the width is decreased by 2 meter the area becomes 119 square meter. Find the dimensions of the original rectangle.
24. Find the range of values of $$x$$ for which $$-x^2 + 3x + 4 > 0$$
25. Find all integral values of $$x$$ for which $$5x - 1 < (x + 1)^2 < 7x - 3$$
26. Find all values of $$x$$ for which inequality $$\frac{8x^2 + 16x - 51}{(2x - 3)(x + 4)} > 3$$ holds.
27. Show that the expression $$\frac{x^2 - 3x + 4}{x^2 + 3x + 4}$$ lies between $$7$$ and $$\frac{1}{7}$$ for real values of $$x$$.
28. If $$x$$ be real, prove that the expression $$\frac{x^2 + 34x - 71}{x^2 + 2x - 7}$$ has no value between $$5$$ and $$9$$.
29. If $$x$$ be real, show that the expression $$\frac{4x^2 + 36x + 9}{12x^2 + 8x + 1}$$ can have any real value.
30. Prove that if $$x$$ is real, the expression $$\frac{(x - a)(x - c)}{(x - b)}$$ is capable of assuming all values if $$a > b > c$$ or $$a < b < c$$.
31. If $$x + y$$ is constant, prove that $$xy$$ is maximum when $$x = y.$$
32. If $$x$$ be real find the maximum value of $$3 - 6x - 8x^2$$ and the corresponding value of $$x.$$
33. Prove that $$\left|\frac{12x}{4x^2 + 9}\right| \le 1$$ for all real values of $$x$$ or the equality being satisfied only if $$|x| = \frac{3}{2}$$
34. Prove that if the equation $$x^2 + 9y^2 - 4x + 3 = 0$$ is satisfied for real values of $$x$$ and $$y, x$$ must lie between $$1$$ and $$3$$ and $$y$$ must lie between $$-\frac{1}{3}$$ and $$\frac{1}{3}$$.
35. Find the value of $$a$$ for which $$x^2 - ax + 1 - 2a^2 > 0$$ for all real values of $$x$$.
36. Determine $$a$$ such that $$x^2 - 11x + a$$ and $$x^2 - 14x + 2a$$ may have a common factor.
37. Find the condition that the expression $$ax^2 + bxy + cy^2$$ and $$a_1x^2 + b_1xy + c_1y^2$$ may have factors $$y - mx$$ and $$my - x$$ respectively.
38. Find the values of $$m$$ for which the expression $$2x^2 + mxy + 3y^2 - 5y - 2$$ can be resolved in two linear factors.
39. If the expression $$ax^2 + by^2 + cz^2 + 2ayz + 2bzx + 2cxy$$ can be resolved into two rational factors, prove that $$a^3 + b^3 + c^3 = 3abc.$$
40. Find the linear factors of $$2x^2 - y^2 - x + xy + 2y - 1$$
41. Show that the expression $$x^2 + 2(a + b + c)x + 3(bc + ca + ab)$$ will be a perfect square if $$a = b = c.$$
42. If $$x$$ is real prove that $$2x^2 - 6x + 9$$ is always positive.
43. Prove that $$8x - 15 - x^2 > 0$$ for limited values of $$x$$ and also find the limits.
44. Find the range of the values of $$x$$ for which $$-x^2 + 5x - 4 > 0.$$
45. Find the range of the values of $$x$$ for which $$x^2 + 6x - 27 > 0.$$
46. Find the solution set of inequation $$\frac{4x}{x^2 + 3}\ge 1, x \in R.$$
47. Find the real values of $$x$$ which satisfy $$x^2 - 3x + 2 > 0$$ and $$x^2 - 3x - 4 \le 0$$
48. If $$x$$ be real and the roots of the equation $$ax^2 + bx + c = 0$$ are imaginary, prove that $$a^2x^2 + abx + ac$$ is always positive.
49. Prove that the expression $$\frac{x^2 - 2x + 4}{x^2 + 2x + 4}$$ lies between $$\frac{1}{3}$$ and $$3$$ for real values of $$x$$.
50. If $$x$$ be real show that $$\frac{2x^2 - 3x + 2}{2x^2 + 3x + 2}$$ lies between $$7$$ and $$\frac{1}{7}$$.