This note facilitates you to prove good questions based on Quadratic Equations. So please use this note. I will try my best to post these types of notes from now.

\(\color{green}\text{Prove the Following :}\)

If the sum of the roots of the equation \(px^2 + qx + r = 0\) be equal to the sum of their squares, show that \(2pr = pq + q^2\)

If the difference of roots of \(x^2 - bx + c = 0\) be same as that of the roots of \(x^2 - cx + b = 0\) then prove that \(b + c = -4\) and \(b = c\)

If the roots of the equation \(\dfrac{1}{(x + p)} + \dfrac{1}{(x + q)} = \dfrac{1}{r}\) are equal in magnitude but opposite in sign, show that \(p + q = 2r\) and product of the roots is equal to \(-\dfrac{1}{2}(p^2 + q^2)\)

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

If the ratio of the roots of the equation \(x^2 + px + q = 0\) be equal to the ratio of roots of the equation \(x^2 + bx + c = 0\), then prove that \(p^2c = b^2q\)

If the sum of the two roots of the equation \(x^3 - px^2 + qx - r = 0\) is zero, then prove that \(pq = r\)

If the equations \(ax^2 + bx + c = 0\) and \(bx^2 + cx + a = 0\) have a common root than prove that \(a^3 + b^3 + c^3 = 3abc\) and \(a = 0\)

If anybody proved any one of these I will mark that question as proved. But if you have another idea to the same question you can post it too. So I will also mention in how many methods the proof is proved. Try your best to prove these.

For more of these see my set Proof Based Notes

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TopNewestEliminate x and y sin(x) + sin(y) = a cos(x) + cos(y) = b tan(x) + tan(y) = c

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1) Let the roots of the equation \(px^2 + qx + r = 0\) be \(a,b\)

Given : \(a + b = a^2 + b^2\)

\(\implies a + b = (a + b)^2 - 2ab\)

\(\implies -\dfrac{q}{p} = \dfrac{q^2}{p^2} - \dfrac{2r}{p}\)

\(\implies - pq = q^2 - 2pr\) (Multiplying all terms by \(p^2\))

\(\implies 2pr = pq + q^2\)

\(\color{green}\text{Hence Proved}\)

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