You are given that x=(2x^2-x+8)y.(1)show,by the method of completing the square that (x-1+y)^2 divided by 4y,=1+2y-63y^2 divided by 16y^2
To show that (x-1+y)^2/4y = 1+2y-63y^2/16y^2 using the method of completing the square, we will follow the steps below:
Step 1: Start with the given equation x = (2x^2 - x + 8)y.
Step 2: Expand the equation on the right-hand side:
2x^2 - x + 8 = (x-1)^2 + 2y - 63y^2/16y^2
Step 3: Divide both sides of the equation by 4y:
(x-1+y)^2 / 4y = (x-1)^2 / 4y + 2y - 63y^2 / (16y^2 * 4y)
Step 4: Simplify the right-hand side:
(x-1+y)^2 / 4y = (x-1)^2 / 4y + 2y - 63y^2 / (64y^3)
Step 5: Reduce the fractions:
(x-1+y)^2 / 4y = (x-1)^2 / 4y + 2y - 63 / 64y
Step 6: Combine the fractions:
(x-1+y)^2 / 4y = [(x-1)^2 + 8y - 63] / 64y
Step 7: Simplify further:
(x-1+y)^2 / 4y = (x^2 - 2x + 1 + 8y - 63) / 64y
(x-1+y)^2 / 4y = (x^2 - 2x - 62 + 8y) / 64y
Step 8: Use the given equation, x = (2x^2 - x + 8)y, to substitute for x:
(x-1+y)^2 / 4y = ((2(2x^2 - x + 8)y) - 2x - 62 + 8y) / 64y
Simplifying further would require additional information about the value of x or y.