1. 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 \(a ≠ 0\).

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. Translate: \(\cfrac{x}{4 - x} = \cfrac{1}{3x}, \ (x \ne 0, \ x \ne 4)\) — If we express this equation in the form of a quadratic equation \(ax^2 + bx + c = 0\) where \(a \ne 0\), then let's determine the coefficient of \(x\).

(a) 1 (b) 2 (c) 3 (d) 4

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

5. 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 \(x^0\).

6. 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 \(2b^2 = 9ac\).

7. 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

8. 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 \(a ≠ 0\).

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

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

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

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

14. If the roots of the quadratic equation \(ax^2+bx+c=0\) are in the ratio \(1:p\), prove that \(\cfrac{(p+1)^2}{p} = \cfrac{b^2}{ac}\).

15. In the quadratic equation \(ax^2 + bx + c = 0\), if one root is three times the other, prove that \(3b^2 = 16ac\).

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. Translate to English: If one root of the quadratic equation \(ax^2 + bx + c = 0\) is double the other, show that \(2b^2 = 9ac\).

18. Express the equation \(x^3 - 4x^2 - x + 1 = (x + 2)^3\) in the standard form of a quadratic equation and write the coefficients of \(x^2\), \(x\), and \(x^0\).

19. 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 \( ax^2 + bx + c = 0 \).

20. If one root of the quadratic equation \(ax^2 + bx + c = 0\) is twice the other, prove that \(2b^2 = 9ac\).

21. If one root of the quadratic equation \(ax^2 + bx + c = 0\) is twice the other, prove that \(2b^2 = 9ac\).

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

23. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1:r\), then show that \((r+1)^2ac = b^2r\).

24. If the quadratic equation \(ax^2 + 7x + b = 0\) has roots \(\cfrac{2}{3}\) and \(-3\), then find the values of \(a\) and \(b\).

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

26. If one root of the quadratic equation \(ax^2 + bx + c = 0\) is twice the other, show that \(2b^2 = 9ac\).

27. If one root of the quadratic equation \(ax^2 + bx + c = 0\) is twice the other, show that \(2b^2 = 9ac\).

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

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

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