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If A, B, C Are Real Numbers Such That Ac ≠ 0, Then Show that at Least One of the Equations Ax2 + Bx + C = 0 and -ax2 + Bx + C = 0 Has Real Roots. - CBSE Class 10 - Mathematics

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Question

If a, b, c are real numbers such that ac ≠ 0, then show that at least one of the equations ax2 + bx + c = 0 and -ax2 + bx + c = 0 has real roots.

Solution

The given equations are

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

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

Roots are simultaneously real

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

Then,

D1 = (b)2 - 4ac

= b2 - 4ac

And

D2 = (b)2 - 4 x (-a) x c

= b2 + 4ac

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

b2 - 4ac ≥ 0             

b2 ≥ 4ac                        ................. (3)

And,

b2 + 4ac ≥ 0                ............... (4)

Now given that a, b, c are real number and ac ≠ 0 as well as from equations (3) and (4) we get

At least one of the given equation has real roots

Hence, proved

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Solution If A, B, C Are Real Numbers Such That Ac ≠ 0, Then Show that at Least One of the Equations Ax2 + Bx + C = 0 and -ax2 + Bx + C = 0 Has Real Roots. Concept: Nature of Roots.
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