trouvez tous les nombres dont le tiers du carré est égal au quart du carré augmenté de 1

x : le nombre

1/3 x^2 = 1/4 (x + 1)^2
1/3 x^2 = 1/4 (x^2 + 2x + 1)
1/3 x^2 - 1/4 x^2 - 2/4x - 1/4 = 0
4/12 x^2 - 3/12 x^2 - 1/2x - 1/4 = 0
1/12 x^2 - 1/2x - 1/4 = 0
1/4 (1/3 x^2 - 2x - 1) = 0

On calcule le discriminant :
Delta = (-2)^2 - 4 * (1/3) * (-1)
Delta = 4 + 4/3
Delta = 12/3 + 4/3
Delta = 16/3 > 0 donc deux solutions

X1 = (2 - V(16/3)) / (2 * 1/3)
X1 = (2 - 4/V3) / (2/3)
X1 = (2 - 4/V3) * (3/2)
X1 = (3 - 6/V3)
X1 = -0,46

X2 = (2 + V(16/3)) / (2 * 1/3)
X2 = (2 + 4/V3) / (2/3)
X2 = (2 + 4/V3) * (3/2)
X2 = (3 + 6/V3)
X2 = 5,45