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.