if y=(x +3)2, then (-2x-6)2 must equal which of the following:
a) -4y2
b) -2y2
c) -4y
d) 4y
given: y =(x + 3)^2
(-2x - 6)^2
((-2)(x+3))^2
distribute the exponent:
(-2)^2 * (x+3)^2
substitute y:
4 * y
which is letter D.
To find the value of (-2x - 6)² in terms of y, we need to substitute the expression for y from the given equation (y = (x + 3)²).
Step 1: Substitute y with (x + 3)² in (-2x - 6)²:
((-2x - 6)² = (-2(x + 3))²
Step 2: Expand the expression (-2(x + 3))²:
((-2(x + 3))² = (-2)² * (x + 3)²
((-2(x + 3))² = 4 * (x + 3)²
Step 3: Simplify the expression 4 * (x + 3)²:
4 * (x + 3)² = 4(x + 3)(x + 3)
4 * (x + 3)² = 4(x² + 3x + 3x + 9)
4 * (x + 3)² = 4(x² + 6x + 9)
4 * (x + 3)² = 4x² + 24x + 36
Step 4: Substitute y with (x + 3)² in the given answer options:
a) -4y² = -4(x + 3)² = -4(x² + 6x + 9) = -4x² - 24x - 36
b) -2y² = -2(x + 3)² = -2(x² + 6x + 9) = -2x² - 12x - 18
c) -4y = -4(x + 3) = -4x - 12
d) 4y = 4(x + 3) = 4x + 12
Comparing our derived expression 4x² + 24x + 36 to the given options, we see that the correct answer is:
d) 4y = 4(x + 3) = 4x + 12.