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

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

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

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

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

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

7. 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 \(2b^2 = 9ac\).

8. Show that if one root of the quadratic equation \( ax^2 + bx + c = 0 \) is double the other, then \( 2a^2 = 9ac \).

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

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

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

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

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

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

15. 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 \(a \ne 0\).

16. If one root of the quadratic equation \(ax^2 + bx + c = 0\) is zero.

(a) \(a=0\) (b) \(b=0\) (c) \(c=0\) (d) None of the above

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

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

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

20. If one root of the equation \[ ax^2 + bx + c = 0 \] is double the other, prove that \[ 2b^2 = 9ac \]

21. 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.

22. 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

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

24. Under what condition will one root of the quadratic equation \(ax^2 + bx + c = 0\) be zero?

(a) \(a=0\) (b) \(b=0\) (c) \(c=0\) (d) None of these

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

26. 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.

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

28. “If the ratio of the roots of the quadratic equation \( ax^2 + bx + c = 0 \) (where \( a \ne 0 \)) is \(1:r\), then show that \(\cfrac{(r+1)^2}{r} = \cfrac{b^2}{ac}\)”

29. If the sum of the roots of the quadratic equation \(x^2 - px + q = 0\) is three times their product, then show that \(2p^2 = 9q\).

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