Question

[Thanh Tu Nguyen]

M = \(\left(x-a\right)\left(x-b\right)+\left(x-b\right)\left(x-c\right)+\left(x-c\right)\left(x-a\right)+x^2\)

Tính M theo a, b, c biết rằng x = \(\dfrac{1}{2}a+\dfrac{1}{2}b+\dfrac{1}{2}c\)

Answer 1

[when the imposter is sus]

Ta có \(x=\dfrac{1}{2}a+\dfrac{1}{2}b+\dfrac{1}{2}c=\dfrac{a+b+c}{2}\)

Suy ra

M = (x - a)(x - b) + (x - b)(x - c) + (x - c)(x - a) + x²

    = x² - ax - bx + ab + x² - bx - cx + bc + x² - ax - cx + ac + x²

    = 4x² - 2ax - 2bx - 2cx + ab + bc + ac

    = (2x)² - 2x(a + b + c) + ab + bc + ac

    = \(\left(2\cdot\dfrac{a+b+c}{2}\right)^2-\left(2\cdot\dfrac{a+b+c}{2}\right)\left(a+b+c\right)+ab+bc+ac\)

    = ab + bc + ac