If x and y are real numbers such that 4x² +4y² −4xy −6y +3 = 0, then the value of 4x +5y is:

The correct option is 7.

To find the value of 4x+5y, we need to determine the values of the real numbers x and y that satisfy the given equation:

4 x 2 + 4 y 2 − 4 x y − 6 y + 3 = 0

We can solve this equation by grouping and completing the square. Let's rewrite the equation by splitting the term 4y2 into y2+3y2:

4 x 2 − 4 x y + y 2 + 3 y 2 − 6 y + 3 = 0

Now, let's simplify each group:
1. The first group 4x2−4xy+y2 is a perfect square trinomial, which can be factored as 2x−y2.
2. For the second group, we can factor out 3 to get 3y2−2y+1, which simplifies to 3y−12.

Substituting these factored forms back into our equation gives:

2 x − y 2 + 3 y − 1 2 = 0

Since x and y are real numbers, the square terms 2x−y2 and y−12 must both be greater than or equal to zero. The only way their sum can equal zero is if each individual squared term is exactly zero:

y − 1 2 = 0 ⇒ y − 1 = 0 ⇒ y = 1

Using the value of y=1 in the first term:

2 x − y 2 = 0 ⇒ 2 x − y = 0 ⇒ 2 x = y ⇒ 2 x = 1 ⇒ x = 1 2

Now, we substitute the values of x=12 and y=1 into the expression 4x+5y:

4 x + 5 y = 4 1 2 + 5 1 = 2 + 5 = 7