Question

Can someone please solve Q32, 37 and 47, they are related to quadratic equations.

Answer 1

Answer:

Q32) we have:

x² - Px + P = x(x - P) + P

but x² - Px + P is divisible of x - P

=> P must be divisible of x - P

=> P = 0

Q37) 3x² + Px + 54 = 0

a = 3 b = P c = 54

=> delta = b² - 4ac = P² - 648

because this equation have 2 solutions => delta > 0

=> P² - 648 > 0

$= > p < - 18 \sqrt{2} \: or \: p > 18 \sqrt{2}$

x1, x2 are 2 solutions of this equation

=> x1 = 2.x2

=> x1 - 2.x2 = 0

Using Viète theorem, we have:

x1 + x2 = -P/3

x1.x2 = 18

but x1 = 2.x2 => 3.x2= -P/3

2.x2² = 18

=> -9.x2 = P

x2² = 9 => x2 = 3 or - 3

=> P = 27 or - 27

Q47) because Px² + 2x³ + qx + c is divided by x+ 1 and x-2

=> P - 2 - q + c = 0

4P + 16 + 2q + c = 0

=> 3p + 3q = - 18

=> p + q = -6