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Standard X

Mathematics

Solving a Quadratic Equation by Completion of Squares Method

If a>b>c an...

Question

If $a > b > c$ and $a^{3} + b^{3} + c^{3} = 3 a b c,$ then the quadratic equations $a x^{2} + b x + c = 0$ has roots which are

both real

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one

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Solution

The correct option is A both real
$a^{3} + b^{3} + c^{3} - 2 a b c = \frac{1}{2} (a + b + c) [{(a - b)}^{2} + {(b - c)}^{2} + {(c - a)}^{2}] = 0$
$[∵ a > b > c$ , so ${(a - b)}^{2} + {(b - c)}^{2} + {(a - c)}^{2} > 0]$
$∴ a + b + c = 0$ ...(i)
Comparing (i) with ${a x}^{2} + b x + c = 0,$ we find $x = 1.$
Also, ${a x}^{2} + b x + c = 0$ for $x \leq \frac{c}{a}$
Therefore, the roots of ${a x}^{2} + b x + c = 0$ are $1$ and $\frac{c}{a} .$

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Similar questions

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If a>b>c and a3+b3+c3=3abc, then the quadratic equations ax2+bx+c=0 has roots which are

The correct option is A both reala3+b3+c3−2abc=12(a+b+c)[(a−b)2+(b−c)2+(c−a)2]=0[∵a>b>c, so(a−b)2+(b−c)2+(a−c)2>0]∴a+b+c=0 ...(i)Comparing (i) with ax2+bx+c=0, we find x=1.Also, ax2+bx+c=0 for x≤caTherefore, the roots of ax2+bx+c=0 are 1 and ca.

If $a > b > c$ and $a^{3} + b^{3} + c^{3} = 3 a b c,$ then the quadratic equations $a x^{2} + b x + c = 0$ has roots which are