Question

1. $y = -2x^2 - 16x - 30$, for $y = -6$

$\frac{-(-16)}{2(-2)} = \frac{16}{-4} = $
graph with x-axis from -10 to 10 and y-axis from -10 to 10, grid lines
points: ________ and ________
$x = $ ____ and $x = $ ____

Explanation:

Step1: Substitute y = -6 into the equation

We have the equation \( y = -2x^2 - 16x - 30 \). Substitute \( y = -6 \) into it:

$$ -6 = -2x^2 - 16x - 30 $$

Step2: Simplify the equation

Add 6 to both sides to set the equation to zero:

$$ 0 = -2x^2 - 16x - 30 + 6 $$

$$ 0 = -2x^2 - 16x - 24 $$

Divide both sides by -2 to simplify:

$$ 0 = x^2 + 8x + 12 $$

Step3: Factor the quadratic equation

Factor \( x^2 + 8x + 12 \). We need two numbers that multiply to 12 and add to 8. Those numbers are 2 and 6. So:

$$ x^2 + 8x + 12 = (x + 2)(x + 6) $$

Step4: Solve for x

Set each factor equal to zero:
\( x + 2 = 0 \) gives \( x = -2 \)
\( x + 6 = 0 \) gives \( x = -6 \)

Step5: Find the points

For \( x = -2 \), \( y = -6 \), so the point is \( (-2, -6) \)
For \( x = -6 \), \( y = -6 \), so the point is \( (-6, -6) \)

Answer:

Points: \((-6, -6)\) and \((-2, -6)\)
\( x = -6 \) and \( x = -2 \)