To solve the equation (y+10)^2 -27=0, we can follow these steps:
1. Expand the square: (y+10)^2 = 27. This can be done by multiplying (y+10) by itself.
(y+10)(y+10) = 27.
2. Apply the FOIL method to expand the expression:
y * y + y * 10 + 10 * y + 10 * 10 = 27.
y^2 + 10y + 10y + 100 = 27.
3. Combine like terms:
y^2 + 20y + 100 = 27.
4. Move the constant term to the other side of the equation:
y^2 + 20y + 100 - 27 = 0.
5. Simplify:
y^2 + 20y + 73 = 0.
Now, we have a quadratic equation in the form of ay^2 + by + c = 0, where a = 1, b = 20, and c = 73.
To find the solutions, we can use the quadratic formula:
y = (-b ± √(b^2 - 4ac))/(2a).
6. Substitute the values of a, b, and c into the formula:
y = (-20 ± √(20^2 - 4 * 1 * 73))/(2 * 1).
7. Simplify:
y = (-20 ± √(400 - 292))/2.
y = (-20 ± √108)/2.
y = (-20 ± √(4 * 27))/2.
y = (-20 ± 2√27)/2.
8. Simplify further:
y = -10 ± √27.
So the two solutions for the equation (y+10)^2 - 27 = 0 are:
y = -10 + √27
y = -10 - √27
Please note that these solutions cannot be simplified further, and they are the exact solutions for the equation.