Question

11 identify the interval(s) on which the function $y = x^2 + 12x + 27$ is positive.
a $(-9, -3)$
b $(-\infty, -9)$ and $(-3, \infty)$
c $(-\infty, 3)$ and $(9, \infty)$
d $(-\infty, -9)$ and $(3, \infty)$

Explanation:

Step1: Factor the quadratic function

To find where \( y = x^2 + 12x + 27 \) is positive, first factor the quadratic. We need two numbers that multiply to \( 27 \) and add to \( 12 \). Those numbers are \( 9 \) and \( 3 \). So, \( x^2 + 12x + 27=(x + 9)(x+ 3) \).

Step2: Find the roots

Set \( y = 0 \), then \( (x + 9)(x + 3)=0 \). Using the zero - product property, we get \( x=-9 \) or \( x = - 3 \). These are the x - intercepts of the parabola.

Step3: Analyze the parabola's shape

The coefficient of \( x^2 \) in the quadratic function \( y=x^2 + 12x + 27 \) is \( a = 1>0 \), so the parabola opens upwards.

Step4: Determine the intervals where \( y>0 \)

For a parabola that opens upwards, the function is positive when \( x < - 9 \) (to the left of the smaller root) or \( x>-3 \) (to the right of the larger root). So the intervals where \( y=x^2 + 12x + 27 \) is positive are \( (-\infty,-9)\) and \( (-3,\infty) \).

Answer:

B. \((-\infty, -9)\) and \((-3, \infty)\)