Find the solutions of the equation \(\displaystyle{\frac{{{a}^{{2}}{\left({x}-{b}\right)}{\left({x}-{c}\right)}}}{{{\left({a}-{b}\right)}{\left({a}-{c}\right)}}}}+{\frac{{{b}^{{2}}{\left({x}-{a}\right)}{\left({x}-{c}\right)}}}{{{\left({b}-{a}\right)}{\left({b}-{c}\right)}}}}+{\frac{{{c}^{{2}}{\left({x}-{a}\right)}{\left({x}-{b}\right)}}}{{{\left({c}-{a}\right)}{\left({c}-{b}\right)}}}}={3}{x}-{2}\)

Ekwadorkadba3

Ekwadorkadba3

Answered question

2022-04-06

Find the solutions of the equation
a2(xb)(xc)(ab)(ac)+b2(xa)(xc)(ba)(bc)+c2(xa)(xb)(ca)(cb)=3x2

Answer & Explanation

StettyNagEragpouj

StettyNagEragpouj

Beginner2022-04-07Added 7 answers

Step 1
Assume a,b,c are fixed distinct (all different) real number values. (Otherwise the left-hand side is undefined.)
Then the left-hand side is a quadratic function in x. Call this function f(x).
We have f(a)=a2, f(b)=b2 and f(c)=c2. Since a quadratic is determined by its values at three points, we must have f(x)=x2 (This expression for f(x) should be independent of the values of a,b, and c, provided, as noted above, that these values are all different.)
So the equation becomes x2=3x2, which you can now easily solve for x.
tempur8x43

tempur8x43

Beginner2022-04-08Added 16 answers

Step 1
Add x2 on both sides , Let expression on LHS be f(x)
Therefore LHS will be a quadratic in x will be become zero three times clearly as
f(a)=f(b)=f(c) Now if a quadratic equation has more than three roots, then all complex numbers will be it's roots and therefore it will identically be equal to zero.
f(x)=0 and therefore since f(x)=3x2x2
3x2x2=0
which is possible for x=1, 2 only

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