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Common Roots

If the equati...

Question

If the equations px² + 2qx + r = 0 and $q x^{2} - 2 \sqrt{p r} x + q = 0$ have real roots, then q² =____________.

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Solution

$Given p x^{2} + 2 q x + r = 0 and q x^{2} - 2 \sqrt{p r} x + q = 0 have real roots for p x^{2} + 2 q x + r = 0 Since roots are real \Rightarrow Discriminant D \geq 0 i . e . b^{2} - 4 a c \geq 0 i . e . {(2 q)}^{2} - 4 (p) (r) \geq 0 i . e . 4 q^{2} - 4 p r \geq 0 i . e . q^{2} \geq p r . . . (i) also for q x^{2} - 2 \sqrt{p r} x + q = 0 D \geq 0 i . e . b^{2} - 4 a c \geq 0 i . e . 4 pr - 4 (q) (q) \geq 0 i . e . p r \geq q^{2} i . e . q^{2} \leq p r . . . (ii) ∴ from (i) and (ii) p r \leq q^{2} \leq p r \Rightarrow q^{2} = p r$

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