find the value of k?

Question

find the value of k such that

y^2 + 10y + k

is a perfect square

Answer 1

y^2 + 10y + k

For this to be a perfect square, k must be the square of a rational number.

This expression is in the form (y + x)^2 = y^2 + 2xy + x^2
Where x^2 = k

In the second term 10 y = 2 * 5 * y
So x = 5
k = 5^2 = 25

Answer 2

To find k you must complete the square.

(x+a)^2 = x^2 + 2ax + a^2

we want to write out y^2 + 10y in the form (y+a)^2

to find a we divide 10 by 2 = 5

so (y+5)^2 = y^2 + 10y + 25 this is not equal to y^2 +10y, so to make it equal we must write:

(y+5)^2 - 25.
so k=25

Another way to think about this is to imagine a square, since it is a square there is the same length on both sides, to get the area y^2, we put a y on each side of the square, to get the 10y we need to put 5 on each side of the square. By putting 5 on each side of the square we create two areas with length 5 and width y adding up to give area 10y, but by doing this we also create an are with l=5 and w=5, so we need to minus this to give the area of y^2 +10y, this is 5^2. k is this number, 25.

hope that helps!

Answer 3

Take 10 and divide by 2 then square it:
10/2=5
5^2=25
k=25

Check:
y^2+10y+25
(y+5)^2

I hope this helps!

Answer 4

Take half of 10, square it and add it to y² + 10 y
The answer is 25. y² + 10y + 25 = (y+5)².
This is the process of completing the square.

Answer 5

y^2+10y+k
=(y)^2+2*y*5+(5)^2-(25-k)
=(y+5)^2-(25-k)
So to make the expression a perfect square (25-k) must be zero or k should be equal to 25

Answer 6

25

Answer 7

(y+x)(y+x) = y^2+2xy+x^2
2xy = 10y
x=5
k=25

k?