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If the Roots of the Equations Ax2 + 2bx + C = 0 and Bx^2-2sqrt(Ac)X+B = 0 Are Simultaneously Real, Then Prove That B2 = Ac. - CBSE Class 10 - Mathematics

Question

If the roots of the equations ax2 + 2bx + c = 0 and bx^2-2sqrt(ac)x+b = 0 are simultaneously real, then prove that b2 = ac.

Solution

The given equations are

ax2 + 2bx + c = 0             ............ (1)

bx^2-2sqrt(ac)x+b = 0 ............. (2)

Roots are simultaneously real

Then prove that b2 = ac

Let D1 and D2 be the discriminants of equation (1) and (2) respectively,

Then,

D1 = (2b)2 - 4ac

= 4b2 - 4ac

And

D_2=(-2sqrt(ac))^2-4xxbxxb

= 4ac - 4b2

Both the given equation will have real roots, if D1 ≥ 0 and D2 ≥ 0

4b2 - 4ac ≥ 0

4b2 ≥ 4ac

b2 ≥ ac                ............... (3)

4ac - 4b2 ≥ 0

4ac ≥ 4b2

ac ≥ b2                          ................... (4)

From equations (3) and (4) we get

b2 = ac

Hence, b2 = ac

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Solution If the Roots of the Equations Ax2 + 2bx + C = 0 and Bx^2-2sqrt(Ac)X+B = 0 Are Simultaneously Real, Then Prove That B2 = Ac. Concept: Nature of Roots.
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