Final answer:
The Zero Product Property is used to solve the quadratic equation y = -x² - 7x - 12 by factoring it into y = -(x + 4)(x + 3) and then setting each factor to zero. The solutions are x = -4 and x = -3.
Step-by-step explanation:
The student is asking how to use the Zero Product Property to find the solutions of the quadratic equation y = -x² - 7x - 12. To apply the Zero Product Property, the equation must be factored into a product of binomials set equal to zero. The given quadratic is a parabola that opens downwards given the negative coefficient of the x² term.
First, we need to factor the quadratic equation:
To factor, we look for two numbers that multiply to give -12 (the constant term) and add to give -7 (the coefficient of the x term). The numbers that satisfy these conditions are -4 and -3. Therefore, we can factor the quadratic as:
Next, we apply the Zero Product Property, which states that if a product of two terms is zero, then at least one of the terms must be zero. We set each factor equal to zero and solve for x:
- x + 4 = 0 → x = -4
- x + 3 = 0 → x = -3
Thus, the solutions to the quadratic equation are x = -4 and x = -3.