1. 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

2. 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

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 \(5x^2 − 2x + 3 = 0\) is a quadratic equation with roots \(α\) and \(β\), find the value of \(\frac{1}{α} + \frac{1}{β}\).

5. If the product of two consecutive positive odd numbers is 143, form the equation and find the two numbers using Sridhar Acharya's formula (the quadratic formula).

6. Five times a positive integer is 3 less than twice the square of that integer. Form the required quadratic equation to find the integer, and then solve the equation to determine its value.

7. Find the value of \(m\) if the roots of the quadratic equation \(4x^2+4(3m+1)x+(m-7)-20=0\) are distinct.

8. If \(\cfrac{1}{3}\) is one root of the quadratic equation \(3x^2-10x+3=0\), determine the other root.

9. Divide 42 into two parts such that one part is equal to the square of the other part. — Form a quadratic equation with one variable from the given statement.

10. The distance between two stations is 300 km. A train travels from the first station to the second at a uniform speed. If the speed of the train had been 5 km/h more, it would have taken 2 hours less to reach the second station. — Form a quadratic equation with one variable from the given statement.

11. In a two-digit number, the unit digit is 6 more than the tens digit, and the product of the digits is 12 less than the number itself. — Form a quadratic equation with one variable from the given statement.

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

13. If \(5x^2 + 2x - 3 = 0\) is a quadratic equation whose roots are \(\alpha\) and \(\beta\), find the value of \(\alpha^3 + \beta^3\).

14. If \(5x^2 + 2x - 3 = 0\) is a quadratic equation with roots \(\alpha\) and \(\beta\), find the value of \(\frac{1}{\alpha} + \frac{1}{\beta}\).

15. If \(5x^2 + 2x - 3 = 0\) is a quadratic equation with roots \(\alpha\) and \(\beta\), find the value of \(\frac{α^2}{β} + \frac{β^2}{α}\).

16. The length of a rectangular field is 36 meters more than its breadth. The area of the field is 460 square meters. Form a quadratic equation in one variable from this statement and determine the coefficients of \(x^2\), \(x\), and \(x^0\).

17. The length of a rectangle is 2 meters more than its breadth, and the area of the rectangle is 24 square meters. Form a quadratic equation in one variable.

18. Find the value of \(k\) such that one of the roots of the quadratic equation \(x^2 + kx + 3 = 0\) is \(1\). Show the calculation.

19. I will solve the quadratic equation \(5x^2 + 23x + 12 = 0\) using an alternative method—that is, by multiplying both sides of the equation by 5 and then finding the roots through the method of completing the square.

20. Got it — sticking strictly to translation. Here's the English version without any extra commentary: If \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( 5x^2 - 3x + 6 = 0 \), then \( \alpha + \beta = -\frac{-3}{5} = \frac{3}{5} \) and \( \alpha\beta = \frac{6}{5} \) \(\therefore \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha\beta} \) \( = \frac{\frac{3}{5}}{\frac{6}{5}} = \frac{3}{5} \times \frac{5}{6} = \frac{1}{2} \) (Answer)

21. In a quadratic equation with one variable, if the constant term (the term with zero power of the variable) is missing, then one root of the equation will always be zero.

(a) 3 (b) 2 (c) 1 (d) 0

22. A specific quadratic equation has equal coefficients for \(x^2\) and the constant term (\(x^0\)) — Show with reasoning that the equation whose roots are the reciprocals of this equation’s roots is identical to the original equation.