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 quadratic equation \(ax^2 + bx + c = 0\) is twice the other, prove 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 equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) is double the other, show that \(2b^2 = 9ac\).

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

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

8. In the equation \( ax^2 + bx + c = 0 \), if one root is twice the other, prove that \( 2b^2 = 9ac \).

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

10. 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.

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

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

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

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

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

17. 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

18. “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 \(\cfrac{(r+1)^2}{r} = \cfrac{b^2}{ac}\)”

19. A quadratic equation in one variable \(x\), with real coefficients, has equal coefficients for \(x^2\) and the constant term. Show with reasoning that any other quadratic equation whose roots are the reciprocals of the original equation's roots must be identical to the original equation.

20. 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.

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