If 3 is a root of the quadratic equation x2 – x + k – 0, find the

1.	If 3 is a root of the quadratic equation x2 – x + k – 0, find the value of p so that the roots of the equation x2 + k (2x + k + 2) + p = 0 are equal.
Answer» Given: 3 is a root of equation x² – x + k = 0 Substitute the value of x = 3 (3)² – (3) + k = 0 9 – 3 + k = 0 k = -6 Now, x² + k (2x + k + 2) + p = 0 x² + (-6)(2x – 6 + 2) + p = 0 x² – 12x + 36 – 12 + p = 0 x² – 12x + (24 + p) = 0 Compare given equation with the general form of quadratic equation, which is ax² + bx + c = 0 a = 1, b = -12, c = 24 + p Find Discriminant: D = b² – 4ac = (-12)² – 4 x 1 x (24 + p) = 144 – 96 – 4p = 48 – 4p Since roots are real and equal, put D = 0 48 – 4p = 0 4p = 48 p = 12 The value of p is 12.

If 3 is a root of the quadratic equation x2 – x + k – 0, find the value of p so that the roots of the equation x2 + k (2x + k + 2) + p = 0 are equal.

Answer»

Given: 3 is a root of equation x² – x + k = 0

Substitute the value of x = 3

(3)² – (3) + k = 0

9 – 3 + k = 0

k = -6

Now, x² + k (2x + k + 2) + p = 0

x² + (-6)(2x – 6 + 2) + p = 0

x² – 12x + 36 – 12 + p = 0

x² – 12x + (24 + p) = 0

Compare given equation with the general form of quadratic equation, which is ax² + bx + c = 0

a = 1, b = -12, c = 24 + p

Find Discriminant:

D = b² – 4ac

= (-12)² – 4 x 1 x (24 + p)

= 144 – 96 – 4p = 48 – 4p

Since roots are real and equal, put D = 0

48 – 4p = 0

4p = 48

p = 12

The value of p is 12.