1. \(\cfrac{x}{4 - x} = \cfrac{1}{3x},\ (x ≠ 0,\ x ≠ 4)\) — Let us express this in the form of a quadratic equation \(ax^2 + bx + c = 0\), where \(a ≠ 0\), and determine the coefficient of \(x\).

2. Translate: Express \((x + 2)^3 = x(x^2 - 1)\) in the form of a quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\), and write down the coefficients of \(x^2\), \(x\), and \(x^0\) (i.e., the constant term).

3. Let’s translate that into English: Express \(3x^2 + 7x + 23 = (x + 4)(x + 3) + 2\) in the form of a quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\).

4. If the roots of the quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\), are in the ratio 1:2, then show that \(2b^2 = 9ac\).

5. Let us express \(3x^2 + 7x + 23 = (x + 4)(x + 3) + 2\) in the form of a quadratic equation \(ax^2 + bx + c = 0\), where \(a ≠ 0\).

6. If the equation \((x+2)^3 = x(x-1)^2\) is expressed in the form of the quadratic equation \(ax^2 + bx + c = 0\) \((a ≠ 0)\), the coefficient of \(x^0\) (the constant term) will be.

(a) -8 (b) -1 (c) 3 (d) 8

7. Write whether the equation \(x - 1 + \frac{1}{x} = 6,\ (x ≠ 0)\) can be expressed in the form \(ax^2 + bx + c = 0\), where \(a, b, c\) are real numbers and \(a ≠ 0\).

8. If the roots of the equation \(ax^2 + bx + c = 0\) are equal (where \(a \ne 0\)) —

(a) \(c=\cfrac{-b}{2a}\) (b) \(c=\cfrac{b}{2a}\) (c) \(c=\cfrac{b^2}{4a}\) (d) \(c=\cfrac{-b^2}{4a}\)

9. If \(\alpha\) and \(\beta\) are the two roots of the equation \(ax^2 + bx + c = 0\), then \(\cfrac{\alpha}{\beta}\) and \(\cfrac{\beta}{\alpha}\) are also roots of a quadratic equation. Determine that quadratic equation.

10. Let us express \((x + 2)^3 = x(x^2 - 1)\) in the form \(ax^2 + bx + c = 0\), where \(a ≠ 0\), and write down the coefficients of \(x^2\), \(x\), and \(x^0\).

11. If the roots of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) are real and unequal, then \(b^2 - 4ac\) will be —

(a) >0 (b) =0 (c) <0 (d) none of the above

12. If the roots of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) are equal, then —

(a) \(c=\cfrac{b}{2a}\) (b) \(c=-\cfrac{b}{2a}\) (c) \(c=- \cfrac{b^2}{4a}\) (d) \(c= \cfrac{b^2}{4a}\)

13. If one root of the quadratic equation \(ax^2 + abcx + bc = 0\) (\(a \ne 0\)) is the reciprocal of the other, then —

(a) abc=1 (b) b=ac (c) bc=1 (d) a=bc

14. If \( b^3 + a^2c + ac^2 = 3abc \), then determine the relationship between the roots of the equation \( ax^2 + bx + c = 0 \), where \( a \ne b \).

15. If the roots of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) are reciprocals of each other, then \(c =\) _____________.

16. The condition for the quadratic nature of the equation \(ax^2+bx+c=0\) is: \(a \neq 0\).

(a) \(a=0\) (b) \(a\ne 0\) (c) \(a=1\) (d) none of the above

17. If in the equation \(ax^2 + bx + c = 0\), \(a = 0\) and \(b \ne 0\), then the equation is a linear equation.

18. If the roots of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) are reciprocal to each other, then \(c =\) _____________

19. If the roots of the quadratic equation \( ax^2 + 2bx - c = 0 \) (where \( a \ne 0 \)) are real and equal, then \( b^2 \) will be ——.

20. The roots of the equation \(ax^2 + bx - c = 0\) (where \(a \ne 0\)) will be equal if...

(a) \(c=\cfrac{-b}{2a}\) (b) \(c=\cfrac{b}{2a}\) (c) \(c=\cfrac{b^2}{4a}\) (d) \(c=\cfrac{-b^2}{4a}\)

21. Write whether the equation \( x + \frac{3}{x} = x^2 \), where \( x \ne 0 \), can be expressed in the form \( ax^2 + bx + c = 0 \), where \( a, b, c \) are real numbers and \( a \ne 0 \).

22. Write whether the equation \( x^2 - 6\sqrt{x} + 2 = 0 \), where \( a, b, c \) are real numbers and \( a \ne 0 \), can be expressed in the form \( ax^2 + bx + c = 0 \).

23. Write whether the equation \( (x - 2)^2 = x^2 - 4x + 4 \), where \( a, b, c \) are real numbers and \( a \ne 0 \), can be expressed in the form \( ax^2 + bx + c = 0 \).

24. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : r\), then show that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]

25. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : s\), then show that \[ \frac{(s + 1)^2}{s} = \frac{b^2}{ac} \]

26. If the equation \(ax^2+bx+c=0\) has \(a=0\) and \(b\ne 0\), then the equation is a _____ equation.

27. The roots of the equation \(ax^2+bx+c=0, (a \ne 0)\) will be reciprocal if:

(a) \(a=b\) (b) \(a=c\) (c) \(b=c\) (d) \(b^2=4ac\)

28. In the quadratic equation \(ax^2+bx+c=0 (a \ne 0)\), if \(b^2 = 4ac\), then the roots will be real and _____.

29. If the roots of the quadratic equation \(ax^2+bx+c=0\) are in the ratio \(1:2\), then prove that \(2b^2=9ac\).

30. If one root of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) is double the other, show that \(2b^2 = 9ac\).