solve y=2x^2+12x+27 using zero product property

To solve the quadratic equation \( y = 2x^2 + 12x + 27 \) using the zero product property, we need to set \( y \) to 0 and factorize the quadratic expression.

1. Set the equation to zero:
   \[
   2x^2 + 12x + 27 = 0
   \]

2. Factorize the quadratic expression. First, we look for two numbers that multiply to \( 2 \times 27 = 54 \) and add up to 12. However, factoring directly might be challenging. Instead, we use the quadratic formula:

   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]

   Here, \( a = 2 \), \( b = 12 \), and \( c = 27 \).

3. Calculate the discriminant:
   \[
   \Delta = b^2 - 4ac = 12^2 - 4 \cdot 2 \cdot 27 = 144 - 216 = -72
   \]

Since the discriminant is negative, the quadratic equation has no real solutions and therefore cannot be factored into real numbers.

Thus, the equation \( 2x^2 + 12x + 27 = 0 \) does not have real roots, and the zero product property cannot be applied here.

Would you like details on solving quadratic equations with complex roots or any further assistance on a related topic?

Here are five related questions you might want to ask:
1. How do you solve a quadratic equation with complex roots?
2. Can you explain the quadratic formula in detail?
3. How do you complete the square for a quadratic equation?
4. What is the significance of the discriminant in a quadratic equation?
5. How do you factorize quadratic expressions with real coefficients?

**Tip:** When the discriminant of a quadratic equation is negative, the solutions are complex numbers. This is an indication that the graph of the quadratic function does not intersect the x-axis.

Learn how to solve the quadratic equation y = 2x^2 + 12x + 27 using the zero product property with a detailed step-by-step solution. This problem involves quadratic equations and complex numbers, highlighting the quadratic formula. Suitable for high school students in Grades 10-12.

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