17. If the roots of the Equation (a-b)x +(b-c)x+(c-a)=0 are equal. Prove that 2a=b+c
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Answer:
Step-by-step explanation:
If the quadratic equation ax²+bx+c=0 that has roots which are equal then its determinant will also be equal to zero.
= (a-b)x²+(b-c)x+(c-a)=0
where, determinant =0
= (b-c)² -4(a-b)(c-a)==0
= b²+c²-2bc-4ac+4a²+4bc-4ab=0
= b²+c²+4a²+4bc-4ac-4ab=0
= b²+c²+(-2a)²+2bc+2c(-2a)+2(-2a)b=0
= (b+c-2a)²=0
= b+c-2a=0
Therefore,
= b+c=2a
Hence proved.
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