1.) To complete the square for the equation 9x^2 - x - 18 + 10 = 0, we first need to isolate the x term on one side of the equation. Adding 8 to both sides, we get:
9x^2 - x - 8 = 0
Next, we divide both sides by 9 to get the coefficient of x^2 to be 1:
x^2 - (1/9)x - (8/9) = 0
Now, we can complete the square by adding and subtracting (1/18)^2 to the left side of the equation:
x^2 - (1/9)x + (1/18)^2 - (8/9) - (1/18)^2 = 0
Simplifying the left side, we get:
(x - 1/18)^2 - 145/324 = 0
2.) To complete the square for the equation 4y^2 + x + 8y + 1 = 0, we first need to isolate the y terms on one side of the equation. Subtracting 1 and rearranging the terms, we get:
4y^2 + 8y = -x - 1
Next, we can complete the square for the y terms by adding and subtracting (8/4)^2 to the left side of the equation:
4(y + 1)^2 - 16 = -x - 1
Simplifying the left side, we get:
4(y + 1)^2 = -x + 15
Finally, we can divide both sides by 4 to get the coefficient of the y^2 term to be 1:
(y + 1)^2 = (-1/4)x + 15/4