# Maths helps with k constants and simultaneous equations?

I need help with the following questions. It would be appreciated if the method and answer were given, so I have a better understanding of how to work out these types of questions.

1.Solve the simultaneous Equations:

x+y=2

4y^2-x^2=11

2. f(x)=x^2+(k+3)x+k

where k is a real constant

(a) Find the discriminant of f(x) in terms of k

(b) Show that the discriminant of f(x) can be expressed in the form (k+a)^2+b, where a and b are integers to be found.

(c) show that, for all values of k, the equation f(x) = 0 has real roots

Thanks,

Awakening

### 1 Answer

- Anonymous8 years agoBest Answer
1.

x + y = 2

(2y)² - x² = 11

Difference of two squares is 11 so they are 25 and 36.

So 2y=6 or -6 and x=5 or -5.

So x = 5 and y = -3 or x = -5 and y = 3

2.

(a) Discriminant = (k+3)²-4k

= k² + 2k + 9

(b) Discriminant = k² + 2k + 1 + 8 = (k+1)² + 8

(c) (k+1)² cannot be less than 0 so (k+1)²+8 is always positive so there are two real roots.