4z^2+13=16z

To solve the equation 4z^2 + 13 = 16z, we need to find the values of z that make the equation true.

First, let's subtract 16z from both sides of the equation to get all the terms on one side:

4z^2 - 16z + 13 = 0

Next, let's move the constant term to the other side of the equation:

4z^2 - 16z = -13

Now, we can factor out a common factor of 4 from the first two terms on the left side:

4(z^2 - 4z) = -13

Next, we can complete the square for the term inside the parentheses by taking half of the coefficient of z (-4) and squaring it (4):

4(z^2 - 4z + 4) = -13 + 4(4)

4(z - 2)^2 = -13 + 16

Simplifying further:

4(z - 2)^2 = 3

Divide both sides of the equation by 4:

(z - 2)^2 = 3/4

Now, take the square root of both sides of the equation to isolate z - 2:

z - 2 = ±√(3/4)

Simplify the square root:

z - 2 = ±√(3)/√(4)

√(4) = 2, so:

z - 2 = ±√(3)/2

Now, let's solve for z:

z = 2 ±√(3)/2

These are the two solutions to the equation 4z^2 + 13 = 16z.