Question

what are the solutions for the quadratic (y = -2(x - 4)(x + 6))?

(x = -6, 4)
(x = -4, 6)
(x = -2, -1)
(x = 1, -2)

Explanation:

🆕 New Concept Discovered: Zero Product Property
Finding where each factor equals zero

Step 1: Set the equation to zero

To find the solutions (also called roots or x-intercepts) of the quadratic equation, we set \( y = 0 \):

$$ -2(x - 4)(x + 6) = 0 $$

Step 2: Apply the Zero Product Property

If a product of factors equals zero, at least one of the individual factors must be zero. Since the constant \(-2\) cannot equal zero, we set each binomial factor to zero:

$$ x - 4 = 0 $$

$$ x + 6 = 0 $$

Step 3: Solve for x

Solve each simple equation:

1. For \( x - 4 = 0 \):

$$ x = 4 $$

1. For \( x + 6 = 0 \):

$$ x = -6 $$

Thus, the solutions are \( x = -6, 4 \).

Answer:

\( x = -6, 4 \) (the first option)