# 47. Quadratic Equations Problems Part 5¶

1. If roots of the equation $$ax^2 + bx + c = 0$$ be $$\frac{k + 1}{k}$$ and $$\frac{k + 2}{k + 1},$$ prove that $$(a + b + c)^2 = b^2 - 4ac.$$

2. If $$f(x) = ax^2 + bx + c,$$ and $$\alpha, \beta$$ be the roots of the equation $$px^2 + qx + r = 0,$$ show that $$f(\alpha)f(\beta) = \frac{(cp - ar)^2 - (bp - aq)(cq - br)}{p^2}$$. Hence or otherwise, show that if $$ax^2 + bx + c = 0$$ and $$px^2 + qx + r = 0$$ have a common root, then $$bp -aq, cp - ar$$ and $$cq - br$$ are in G. P.

3. If $$a(p + q)^2 + 2pbq + c = 0$$ and $$a(p + r)^2 + 2bpr + c = 0,$$ then show that $$qr = p^2 + \frac{c}{a}.$$

4. If $$\alpha, \beta$$ are the roots of the equation $$x^2 - p(x + 1) - c = 0,$$ show that $$(\alpha + 1)(\beta + 1) = 1 - c.$$ Hence, prove that $$\frac{\alpha^2 + 2\alpha + 1}{\alpha^2 + 2\alpha + c} + \frac{\beta^2 + 2\beta + 1}{\beta^2 + 2\beta + c} = 1$$

5. If $$\alpha, \beta$$ be the roots of the equation $$x^2 + px + q = 0$$ and $$x^{2n} + p^nx^n + q^n = 0,$$ where $$n$$ is an even integer, prove that $$\frac{\alpha}{\beta}, \frac{\beta}{\alpha}$$ are the roots of the equation $$x^2 + 1 + (x + 1)^n = 0.$$

6. If $$\alpha, \beta$$ are the roots of the equation $$x^2 + px + q = 0$$ and also of equation $$x^{2n} + p^nx^n + q^n = 0$$ and if $$\frac{\alpha}{\beta}, \frac{\beta}{\alpha}$$ are the roots of the equation $$x^n + 1 + (x + 1)^n = 0,$$ then prove that $$n$$ must be an even integer.

7. If the roots of the equation $$x^2 - ax + b = 0$$ be real and differ by less than $$c,$$ then show that $$b$$ must lie between $$\frac{a^2 - c^2}{4}$$ and $$\frac{a^2}{4}$$

8. Let $$a, b$$ and $$c$$ be integers with $$a > 1,$$ and let $$p$$ be a prime number. Show that if $$ax^2 + bx + c$$ is equal to $$p$$ for two distinct integral values of $$x,$$ then it cannot be equal to $$2p$$ for any integral value of $$x$$.

9. If $$\alpha$$ and $$\beta$$ are the roots of equation $$x^2 + px + q = 0$$ and $$\alpha^4, \beta^4$$ are the roots of the equation $$x^2 - rx + s = 0,$$ show that the equation $$x^2 - 4qx + 2q^2 - r = 0$$ has real roots.

10. If $$\alpha, \beta$$ are the roots of the equation $$ax^2 + bx + c = 0,$$ and $$\alpha_1, -\beta$$ are those of equation $$a_1x^2 + b_1x + c_1 = 0,$$ show that $$\alpha, \alpha_1$$ are the roots of the equation

$\frac{x^2}{\frac{b}{a} + \frac{b_1}{a_1}} + x + \frac{1}{\frac{b}{c} + \frac{b_1}{c_1}} = 0$
11. How many quadratic equations are possible which remains unchanged when its roots are squared?

12. If $$a, b, c$$ are in G. P. then show that the equations $$ax^2 + 2bx + c = 0$$ and $$dx^2 + 2ex + f = 0$$ have a common root if $$\frac{a}{d}, \frac{b}{e}, \frac{c}{f}$$ are in H. P.

13. If the three equations $$x^2 + ax + 12 = 0, x^2 + bx + 15 = 0$$ and $$x^2 + (a + b)x + 36 = 0$$ have a common positive root, find $$a, b$$ and the roots of the equation.

14. If $$m(ax^2 + 2bx + c) + px^2 + 2qx + r$$ can be expressed in the form of $$n(x + k)^2,$$ then show that $$(ak - b)(qk - r) = (pk - q)(bk - c).$$

15. The real numbers $$x_1, x_2, x_3$$ satisfying the equation $$x^3 - x^2 + \beta x + \gamma = 0$$ are in A. P. Find the intervals in which $$\beta$$ and $$\gamma$$ must lie.

16. If equations $$x^3 + 3px^2 + 3qx + r = 0$$ and $$x^2 + 2px + q = 0$$ have a common root, show that $$4(p^2 - q)(q^2 - pr) = (pq - r)^2$$

17. If $$c \ne 0$$ and the equations $$x^3 + 2ax^2 + 3bx + c = 0$$ and $$x^3 + ax^2 + 2bx = 0$$ have a common root, show that $$(c - 2ab)^2 = (2b^2 - ac)(a^2 - b)$$

18. If equation $$x^3 + ax + b = 0$$ have only real roots, then prove that $$4a^3 + 27b^2 \le 0.$$

19. Let $$\alpha$$ be a root of $$ax^2 + bc + c = 0$$ and $$\beta$$ be a root of $$-ax^2 + bx + c = 0.$$ Show that there exists a root of the equation $$\frac{a}{2}x^2 + bx + c = 0$$ that lie between $$\alpha$$ and :math:beta` or $$\beta$$ and $$\alpha$$ as the case may be($$\alpha, \beta \ne 0$$).

20. If $$a, b, c \in R, a \ne 0$$ and the quadratic equation $$ax^2 + bx + c = 0$$ has no real root, then show that $$(a + b + c)c > 0.$$

21. If $$a < b < c < d,$$ then show that the quadratic equation $$(x - a)(x - c) + \lambda (x - b)(x - d) = 0$$ has real roots for all real values of $$\lambda$$.

22. If $$2a + 3b + 6c = 0, (a, b, c \in R)$$ then show that the equation $$ax^2 + bx + c = 0$$ has at least one root between $$0$$ and $$2.$$

23. If $$a, b, c$$ be non-zero real numbers such that $$\int_0^1 (1 + \cos^8 x)(ax^2 + bx + c)dx = \int_0^2 (1 + \cos^8 x)(ax^2 + bx + c)dx$$ show that equation $$ax^2 + bx + c = 0$$ has at least on real root between $$1$$ and $$2$$.

24. Let $$f(x) = ax^2 + bx + c,$$ where $$a, b, c \in R$$ and $$a \ne 0.$$ If $$f(x) = x$$ has non-real roots, show that equation $$f(f(x)) = x$$ has all non-real roots.

25. Let $$a, b, c$$ be positive integers and consider all the quadratic equations of the form $$ax^2 - bx + c = 0$$ which have two distinct real roots in $$]0, 1[$$. Find the least positive integers $$a$$ and $$b$$ for which such a quadratic equation exist.

26. If equation $$ax^2 - bx + c = 0$$ have two distinct real roots in $$(0, 1), a, b, c \in N,$$ then prove that $$\log_5(abc)\ge 2$$

27. If equation $$ax^2 + bx + 6 = 0$$ does not have two distinct real roots, then find the least value of $$3a + b$$.

28. If equation $$2x^3 + ax^2 + bx + 4 = 0$$ has $$3$$ real roots, where $$a, b > 0,$$ show that $$a + b > -6$$.

29. Show that equation $$x^3 + 2x^2 + x + 5 = 0$$ has only one real root $$\alpha$$ such that $$[\alpha] = -3,$$ where $$[x]$$ denotes the integral part of $$x.$$

30. Solve $$(x^2 + 2)^2 + 8x^2 = 6x(x^2 + 2)$$

31. Solve $$3x^3 = (x^2 + \sqrt{18}x + \sqrt{32})(x^2 - \sqrt{18}x - \sqrt{32}) - 4x^2$$

32. Solve $$(15 + 4\sqrt{14})^t + (15 - 4\sqrt{14})^t = 30$$ where $$t = x^2 - 2|x|$$

33. For $$a \le 0,$$ determine all the roots of the equation $$x^2 - 2a|x - a| - 3a^2 = 0$$

34. Find all solutions of equation $$|x^2 - x - 6| = x + 2,$$ where $$x$$ is a real number.

35. Solve the equation $$2^{|x + 2|} - |2^{x + 1} - 1| = 2^{x + 1} + 1$$

36. Solve $$3^x + 4^x + 5^x = 6^x$$

37. Solve $$(\sqrt{2 + \sqrt{3}})^x + (\sqrt{2 - \sqrt{3}})^x = 2^x$$

38. Let $$\{x\}$$ and $$[x]$$ denote the fractional and integral part of a real number $$x$$ respectively. Solve $$4\{x\} = x + [x]$$

39. For the same notation as previous problem solve $$[x]^2 = x(x - [x])$$

40. Solve $$x^3 - y^3 = 127, x^2y - xy^2 = 42$$

41. Solve the system of equations: $$x - 2y + z = 0, 4x - y - 3z = 0, x^2 - 2xy + 3xz = 14$$

42. Solve $$x^4 + y^4 = 82, x+ y = 4$$

43. Solve $$\sqrt{a(2^x - 2) + 1} = 1 - 2^x, x \in R$$

44. If $$x$$ is an integer, find the integral values of $$m$$ satisfying the equation $$(x - 5)(x + m) + 2 = 0$$

45. Find the positive solutions of the system of equations $$x^{x + y} = y^n$$ and $$y^{x + y}= x^{2n}y^n$$ where $$n > 0$$

46. Solve the equation $$(144)^{|x|} - 2(12)^{|x|} + a = 0$$ for every value of the parameter $$a.$$

47. If $$m$$ and $$n$$ are odd integers show that the equation $$x^2 + 2mx + 2n = 0$$ cannot have rational roots.

48. If $$f(x) = ax^3 + bx^2 + cx + d$$ has local extrema at two points of opposite sign, then prove that the roots of the quadratic equation $$ax^2 + bx + c = 0$$ are real and distinct.

49. If $$a$$ and $$b$$ are real, $$b\ne 0,$$ prove that the roots of the quadratic equation $$\frac{(x - a)(ax - 1)}{x^2 - 1} = b,$$ can never be equal.

50. If $$n$$ and $$r$$ are positive integers such that $$0 < r < n,$$ then show that the roots of the quadratic equation $$^nC_rx^2 + 2~^nC_{r + 1}x + ^nC_{r + 2} = 0$$ are real.