# 51. Quadratic Equations Problems Part 7¶

1. Find the integral part of the greatest root of equation $$x^3 - 10x^2 - 11x - 100 = 0$$

2. If $$n\in N, a_0, a_1, a_2, ..., a_n \in I$$ and $$a_n$$ and $$a_0 + a_1 + ... + a_n$$ are odd numbers, show that equation $$a_0x^n + a_1x^{n - 1} + a_2x^{n - 1} + a_{n - 1}x + a_n = 0$$ cannot have integral roots.

3. If the cubic equation $$f(x) = 0$$ has three real roots $$\alpha, \beta, \gamma$$ such that $$\alpha < \beta < \gamma,$$ show that the equation $$f(x) + f'(x) + f"(x) = 0$$ has a root between $$\alpha$$ and $$\gamma.$$

4. Find the values of $$a$$ for which all the roots of the equation $$x^4 - 4x^3 - 8x^2 + a = 0$$ are real.

5. If the equation $$ax^2 - bx + c = 0$$ has two distinct real roots between $$1$$ and $$2$$ where $$a, b, c \in N,$$ show that $$a \ge 5$$ and $$b \ge 11.$$

6. Show that the equation $$(x - 1)^5 + (x + 2)^7 + (7x - 5)^9 = 10$$ has exactly one real root.

7. Find the value of $$\tan(\theta + \phi)$$ and $$\cot(\theta - \phi)$$ where $$\tan\theta$$ and $$\tan\phi$$ are respectively actual and extraneous root of the equation $$\sqrt{2x + 6} - \sqrt{x + 2} = 3$$

8. Solve $$|x + 1| - |x| + 3|x - 1| - 2|x - 2| = x + 2$$

9. Solve $$2^{|x + 1|} - 2^x = |2^x - 1| + 1$$

10. Solve $$|x^2 - 2x| + y = 1, x^2 + |y| = 1$$

11. Solve $$|x^2 - 4x + 3| + 2x + 5 = 0$$

12. Solve $$x^2 + \frac{9x^2}{(x + 3)^2} = 27$$

13. Solve $$\frac{1}{[x]} + \frac{1}{[2x]} = \{x\} + \frac{1}{3},$$ where $$[x]$$ denotes the integral part of $$x$$ and $$\{x\} = x - [x]$$

14. Solve the $$x^{\frac{2}{3}\left[(\log_2 x)^2 + \log_2 x - \frac{5}{4}\right]} = \sqrt{2}$$ for $$x$$

15. Find all the real solution of the equation $$3x^2 - 8[x] + 1 = 0,$$ where $$[x]$$ demotes the greatest integer function.

16. If $$t > 1$$ solve the equation $$(t + \sqrt{t^2 - 1})^{x^2 - 2x} + (t - \sqrt{t^2 - 1})^{x^2 - 2x} = 2t$$

17. Prove that $$2x^4 + 1402 - y^4 = 0$$ has no integral solution.

18. If $$x \in R$$ and $$a_1, a_2, ..., a_n \in R,$$ then find the value of $$x$$ for which $$\sum_{i = 1}^n (x - a_i)^2$$ is least.

19. Let there be a quotient of two natural numbers in which the denominator is one less than the square of the numerator. If we add two to both numerator and denominator, the quotient will exceed $$\frac{1}{3},$$ and if we subtract $$3$$ from both numerator and denominator, the quotient will be between $$0$$ and $$\frac{1}{10}.$$ Determine the quotient.

20. Let $$f(x)$$ be a quadratic expression which is positive for all real $$x.$$ If $$g(x) = f(x) + f'(x) + f"(x),$$ then for all real $$x,$$ show that $$g(x) > 0.$$

21. By considering the quadratic equation $$f(x) = (a_1x + b_1)^2 + (a_2x + b_2)^2 + ... + (a_nx + b_n)^2$$ prove the inequality $$(a_1b_1 + a_2b_2 + ... + a_nb_n)^2 \le (a_1^2 + a_2^2 + ... + a_n^2)(b_1^2 + b_2^2 + ... + b_n^2)$$

22. Find all real values of $$m$$ for which the equation $$x(x + 1)(x + m)(x + m + 1) = m^2$$ has four real roots.

23. Find all real values of $$a$$ for which the equation $$x^4 + (a - 1)x^3 + x^2 + (a - 1)x + 1 = 0$$ possesses at least two distinct negative roots.

24. Find the real values of the parameter $$a$$ for which the equation $$x^4 + 2ax^3 + x^2 + 2ax + 1 = 0$$ has at least two distinct negative roots.

25. If $$a, b, c \in R$$ and $$a\ne 0,$$ solve the following system of equations in $$n$$ unknowns $$x_1, x_2, ..., x_n$$

\begin{align}\begin{aligned}ax_1^2 + bx_1 + c = x_2\\ax_2^2 + bx_2 + c = x_3\\...\\ax_n^2 + bx_n + c = x_1\end{aligned}\end{align}

when (i) $$(b - 1)^2 < 4ac$$ (ii) $$(b - 1)^2 = 4ac$$ (iii) $$(b - 1)^2 > 4ac$$

26. Solve the inequality $$\log_x\left(x^2 - \frac{3}{16}\right) > 4$$

27. Find the values of $$m$$ for which every solution of inequality $$\log_{\frac{1}{2}}x^2 \ge \log_{\frac{1}{2}}(x + 2)$$ is a solution of the inequality $$49x^2 - 4m^4 \le 0.$$

28. Find the values of parameter $$a$$ for which $$1 + \log_2\left(2x^2 + 2x + \frac{7}{2}\right) \ge \log_2(ax^2 + a)$$ is satisfied by at least one real $$x$$.

29. Prove that the minimum value of $$\frac{(x + a)(x + b)}{x + c}, x > -c$$ is $$(\sqrt{a - c} + \sqrt{b - c})^2.$$

30. If $$x, a, b$$ are real, prove that $$4(a - x)(x - a + \sqrt{a^2 + b^2}) \ngtr a^2 + b^2$$

31. If $$\beta$$ is such that $$\sin2\beta \ne 0,$$ show that for real $$x$$ the expression $$\frac{x^2 + 2x\cos2\alpha + 1}{x^2 + 2x\cos2\beta + 1}$$ always lies between $$\frac{\cos^2\alpha}{\cos^2\beta}$$ and $$\frac{\sin^2\alpha}{\sin^2\beta}.$$

32. Show that for all real values of $$x,$$ the expression $$\frac{2a(x - 1)\sin^2\alpha}{x^2 - \sin\alpha}$$ cannot lie between $$2a\sin^2\frac{\alpha}{2}$$ and $$2a\cos^2\frac{\alpha}{2}.$$

33. Show that the expression $$\tan(x + \alpha)/\tan(x - \alpha)$$ cannot lie between $$\tan^2\left(\frac{\pi}{4} - \alpha\right)$$ and $$\tan^2\left(\frac{\pi}{4} + \alpha\right).$$

34. Prove that for real values of $$x$$ the expression $$\frac{ax^2 + 3x -4}{3x - 4x^2 + a}$$ may have any value provided $$a$$ lies between $$1$$ and $$7.$$

35. Prove that the expression $$\frac{(ax - b)(dx - c)}{(bx - a)(cx - d)}$$ will take any real values when $$x$$ is real provided $$a^2 - b^2$$ and $$c^2 - d^2$$ have the same sign.

36. Find the values of $$m$$ for which both roots of the equation $$x^2 - mx + 1 = 0$$ are less than unity.

37. Find the set of values of $$m$$ for which both roots of equation $$x^2 - mx + 1 = 0$$ are greater than unity.

38. Find the values of $$m$$ for which one root of the equation $$x^2 - mx + 1 = 0$$ is less than unity and the other root is greater than unity.

39. For what values of $$k$$ both the roots of the equation $$x^2 + 2(k - 3)x + 9 = 0$$ lie between $$-6$$ and $$1.$$

40. At what values of $$m$$ is one of the roots of the equation $$(2m + 1)x^2 - mx + m - 2 = 0$$ is greater than unity and the other smaller than unity.

41. Find the values of $$m$$ for which both roots of the equation $$x^2 - 2mx + m^2 - 1 = 0$$ lie between $$-2$$ and $$4.$$

42. Find the values of $$a$$ for which the inequality $$x^2 + ax + a^2 + 6a < 0$$ is satisfied for all $$x \in (1, 2).$$

43. Find the values of the parameter $$a$$ for which there is at least one $$x$$ satisfying the conditions $$x^2 + \left(1 - \frac{3}{2}a\right)x + \frac{a^2}{2} - \frac{a}{2} < 0$$ for $$x = a^2 - \frac{1}{2}.$$

44. Find all the values of the parameter $$a$$ for which the inequality $$4^x - a2^{x} - a + 3 \le 0$$ is satisfied by one real $$x.$$

45. Find all the values of the parameter $$a$$ for which the inequality $$ay^x + 4(a - 1)3^x + a > 1$$ is satisfied for all real values of $$x.$$

46. Find the values of $$a$$ for which the inequality $$3 - |x - a| > x^2$$ is satisfied by at least one negative $$x.$$

47. Find all the real values of $$x$$ such that $$\frac{2x - 1}{2x^3 + 3x^2 + x} > 0$$

48. Find the values of $$x$$ which satisfy the inequality $$\frac{x - 2}{x + 2} > \frac{2x - 3}{4x - 1}$$

49. Find the set of real values of $$x$$ for which $$x^2 + 8|x + 4| - 48 > 0$$

50. Find the values of $$x$$ such that $$\log_{10}(x^2 - 2x - 2) \le 0$$