If x,y are the zeroes of the polynomial p(x)=2x^2 +5x+k satisfying the relation x^2+y^2+ xy then value of k?
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X and Y are the zeros of the given polynomial P(x) = 2x² + 5x + K .
hence, sum of roots = - coefficient of x/coefficient of x²
X + Y = - (5/2) ----------(1)
also product of roots = constant/coefficient of x²
XY = K/2 -------(2)
now,
A/C to question,
X and Y follow the relation X² + Y² + XY =0
hence,
X² + Y² + XY = (X + Y)² -2XY + XY = 0
(X + Y)² - XY = 0
(X + Y)² = XY
now, put equations (1) and (2)
{-5/2}² = k/2
25/4 = k/2
k = 25/2
hence, sum of roots = - coefficient of x/coefficient of x²
X + Y = - (5/2) ----------(1)
also product of roots = constant/coefficient of x²
XY = K/2 -------(2)
now,
A/C to question,
X and Y follow the relation X² + Y² + XY =0
hence,
X² + Y² + XY = (X + Y)² -2XY + XY = 0
(X + Y)² - XY = 0
(X + Y)² = XY
now, put equations (1) and (2)
{-5/2}² = k/2
25/4 = k/2
k = 25/2
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