What is the correct standard form of the equation
√(x+4)2+(y−2)2
explain each step
The given expression is √(x+4)2+(y−2)2.
To write it in standard form, start by expanding the square terms:
(x+4)2 = (x+4)(x+4) = x2 + 8x + 16
(y−2)2 = (y−2)(y−2) = y2 − 4y + 4
So the expression becomes:
√(x2 + 8x + 16 + y2 − 4y + 4)
Combining like terms inside the square root:
√(x2 + y2 + 8x − 4y + 16 + 4)
Simplifying further:
√(x2 + y2 + 8x − 4y + 20)
This is the expression in standard form.
To find the correct standard form of the equation √(x+4)² + (y−2)², you need to simplify and rewrite in a format that represents a specific geometric shape, which is a circle.
Step 1: Square each term inside the square root to eliminate the square root symbol:
(x+4)² + (y−2)²
Step 2: Expand the squared terms:
(x+4)(x+4) + (y−2)(y−2)
Multiplying both terms gives us:
(x+4)(x+4) = x(x+4) + 4(x+4) = x² + 4x + 4x + 16 = x² + 8x + 16
Using the same process for the (y-2)(y-2) term, we get:
(y−2)(y−2) = y(y-2) - 2(y-2) = y² - 2y - 2y + 4 = y² - 4y + 4
Step 3: Combine the expanded terms:
(x² + 8x + 16) + (y² - 4y + 4)
Step 4: Rearrange the terms by reordering and grouping similar terms together:
(x² + y²) + (8x - 4y) + (16 + 4)
Simplifying further gives you:
x² + y² + 8x - 4y + 20
Therefore, the correct standard form of the equation √(x+4)² + (y−2)² is:
x² + y² + 8x - 4y + 20