If the roots of the equations ax2 + 2bx + c = 0 and bx2 – 2\(\sqrt{acx}\)+ b = 0 are simultaneously real then prove that b2 = ac
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If the roots of the equations ax2 + 2bx + c = 0 and bx2 – 2\(\sqrt{acx}\)+ b = 0 are simultaneously real then prove that b2 = ac
If the roots of the equations ax2 + 2bx + c = 0 and bx2 – 2\(\sqrt{acx}\)+ b = 0 are simultaneously real then prove that b2 = ac
It is given that the roots of the equations ax2 + 2bx + c = 0 are real.
∴ D1 = (2b)2 – 4 x a x c ≥ 0
⇒ 4(b2 – ac) ≥ 0
⇒ b2 – ac ≥ 0………(i)
Also, the roots of the equation bx2 – 2\(\sqrt{acx}\) + b = 0 are equal
∴ D1 = (-2\(\sqrt{ac}\))2 – 4 x b x b ≥ 0
⇒ 4(ac – b2) ≥ 0
⇒ -4(b2 – ac) ≥ 0
⇒ b2 – ac ≥ 0……..(2)
The roots of the given equations are simultaneously real if (1) and (2) holds true together. This is possible if
b2 – ac = 0
⇒ b2 = ac