Use the zero product property to find the solutions to the equation $(x+2)(x+3)=12$.
The goal is to solve the equation \((x+2)(x+3) = 12\) using the zero product property. A quadratic equation is typically solved by first bringing all terms to one side of the equation to set it equal to zero, and then factoring the quadratic expression. The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero, allowing us to find the solutions for \(x\).
First, we expand the left side of the equation:
\[(x+2)(x+3) = x \times x + 3x + 2x + 6 = x^2 + 5x + 6.\]
Next, set the entire equation equal to zero by subtracting 12 from both sides:
\[x^2 + 5x + 6 = 12\]
\[x^2 + 5x + 6 - 12 = 0\]
\[x^2 + 5x - 6 = 0.\]
Now we have a quadratic equation in the standard form \(ax^2 + bx + c = 0\).
To factor the quadratic expression \(x^2 + 5x - 6\), we look for two numbers that multiply to \(-6\) and add up to \(5\). These two numbers are \(6\) and \(-1\), because \(6 \times (-1) = -6\) and \(6 + (-1) = 5\).
Thus, the expression can be written as:
\[(x + 6)(x - 1) = 0.\]
Using the zero product property, we set each factor equal to zero:
\[x + 6 = 0 \quad \text{or} \quad x - 1 = 0.\]
Solving these simple linear equations gives:
\[x = -6 \quad \text{or} \quad x = 1.\]
Comparing these solutions to the given options, we find that they match option (d).
Quadratic equations appear frequently in physics and engineering, such as determining the path of a projectile or optimizing the dimensions of a rectangular area for maximum capacity. For example, if you are designing a garden and know the area (\(12 \text{ m}^2\)) and the relationship between the dimensions, like length being 2 meters more than one side and width being 3 meters more than that same side, you can set up the equation \((x+2)(x+3) = 12\) to find the correct dimensions needed for your project. Solving these equations helps us translate real-world constraints into precise mathematical solutions.