Pls I need it like now 1. for What values of
the Constant k
does the equation
have equal roots?

2. find the Positive value of "C" if the
expression
is a perfect
Square

Problem 1

The discriminant formula is

d = b^2 - 4ac

from the original expression given to us, it is in the form ax^2+bx+c with

a = k-1

b = -k

c = -k

So we have a discriminant of

d = b^2 - 4ac

d = (-k)^2 - 4(k-1)(-k)

d = k^2 + 4k(k-1)

d = k^2 + 4k^2 - 4k

d = 5k^2 - 4k

Set this equal to 0 and solve for k. We set d equal to zero because a discriminant of 0 means we have two repeated roots.

d = 0

5k^2 - 4k = 0

k(5k - 4) = 0

k = 0 or 5k-4 = 0

k = 0 or 5k = 4

k = 0 or k = 4/5

Problem 2

For this problem, I'll replace every c with k

Also, I'll replace every y with x

The expression turns into (2k+3)x^2-6x+4-k

We'll use the same idea as problem 1. Match it with ax^2+bx+c to find

a = 2k+3

b = -6

c = 4-k

the discriminant is

d = b^2 - 4ac

d = (-6)^2 - 4(2k+3)(4-k)

d = 36 - 4(-2k^2 + 5k + 12)

d = 36 + 8k^2 - 20k - 48

d = 8k^2 - 20k - 12

Set this equal to zero and solve for k

8k^2 - 20k - 12 = 0

4(2k^2 - 5k - 3) = 0

2k^2 - 5k - 3 = 0

2k^2 - 6k + k - 3 = 0

(2k^2-6k) + (k-3) = 0

2k(k-3) + 1(k-3) = 0

(2k+1)(k-3) = 0

2k+1 = 0 or k-3 = 0

2k = -1 or k = 3

k = -1/2 or k = 3

We ignore k = -1/2 as the instructions state the value of c (which I changed to k) is positive.

h is the x-coordinate of the vertex

step-by-step explanation:

we have

in this function

the point represent the vertex of the function

so

h is the x-coordinate of the vertex

k is the y-coordinate of the vertex