1. If the roots of the quadratic equation \((a^2 + b^2)x^2 + 2(ac + bd)x + (c^2 + d^2) = 0\) are equal, then prove that \(\cfrac{a}{b} = \cfrac{c}{d}\).

2. **"If the roots of the quadratic equation \((a^2 + b^2)x^2 - 2(ac + bd)x + (c^2 + d^2) = 0\) are equal, then prove that \(\cfrac{a}{b} = \cfrac{c}{d}\)."**

3. If the roots of the quadratic equation \[ (a^2 + b^2)x^2 + 2(ac + bd)x + (c^2 + d^2) = 0 \] are equal, prove that \(\frac{a}{b} = \frac{c}{d}\).

4. If the roots of the quadratic equation \((1+m^2)x^2 + 2mcx + (c^2 - a^2) = 0\) are real and equal, prove that \(c^2 = a^2(1+m^2)\).

5. If the roots of the quadratic equation \((1 + m^2)x^2 + 2mcx + (c^2 − a^2) = 0\) are real and equal, show that \(c^2 = a^2(1 + m^2)\).

6. If the roots of the quadratic equation \(2(a^2 + b^2)x + 2(a + b)x + 1 = 0\) are equal, prove that \(a = b\).

7. If the roots of the quadratic equation \((b - c)x^2 + (c - a)x + (a - b) = 0\) are equal, prove that \(2b = a + c\).

8. Prove that the quadratic equation \(x^2(a^2+b^2) + 2(ac+bd)x + (c^2+d^2) = 0\) has no real roots when \(ad \ne bc\).

9. If the roots of the quadratic equation \((b-c)x^2+(c-a)x+(a-b)=0\) are equal, prove that \(2b = a + c\).

10. If the roots of the quadratic equation \((b - c)x^2 + (c - a)x + (a - b) = 0\) are equal, prove that \(2b = a + c\).

11. Translate of your statement in English: If the roots of the equation \((b - c)x^2 + (c - a)x + (a - b) = 0\) are equal, then prove that: \(a + c = 2b\).

12. Prove that \(ps=qr\) if the roots of the quadratic equation \((p^2+q^2)x^2-2(pr+qs)x+(r^2+s^2)=0\) are equal.

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

14. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2+7x+3=0\), prove that: \[ \alpha^3+\beta^3+7(\alpha^2+\beta^2)+3(\alpha+\beta)=0 \]

15. If the roots of \((a^2-bc)x^2+2(b^2-ca)x+(c^2-ab)=0\) are equal, show that \(b=0\) or \(a^3+b^3+c^3=3abc\).

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

17. Let us prove that the quadratic equation \[ x^2(a^2 + b^2) + 2(ac + bd)x + (c^2 + d^2) = 0 \] has no real roots when \(ad \ne bc\).

18. If the roots of the equation \(x^2 + (p - 3)x + p = 0\) are real and equal, then prove—without solving—that the value of \(p\) will be either \(1\) or \(9\).

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

20. If the roots of the quadratic equation \(ax^2+bx+c=0\) are real and unequal, the value of \(b^2-4ac\) will be:

(a) >0 (b) <0 (c) 0 (d) None of these

21. If the roots of the quadratic equation \(ax^2 + 2bx + c = 0\) \((a ≠ 0)\) are real and equal, then \(b^2 =\) _____ .

22. What should be the value of \(m\) so that the roots of the quadratic equation \(4x^2 + 4(3m - 1)x + (m + 7) = 0\) are reciprocals of each other?

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

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

25. What should be the value of \(k\) so that the roots of the quadratic equation \(9x^2 + 3kx + 4 = 0\) are real and equal?

26. If the roots of the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) are equal, prove that \(\cfrac{1}{a} + \cfrac{1}{c} = \cfrac{2}{b}\).

27. 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.

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

29. What should be the value of \(m\) so that the roots of the quadratic equation \(4x^2 + 4(3m - 1)x + (m + 7) = 0\) are reciprocals of each other?

30. 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.