- Group the variables \(x^2\), \(y^2\), and constant terms in the original expression.
- Equate the coefficients of the grouped terms to the coefficients of the simplified expression \(9x^2 - y^2 + 9\).
- Solve the resulting equations \(3 + C_1 = 9\) and \(9 + C_2 = -1\) to find the missing terms \(+6x^2\) and \(-10y^2\).
- The correct option is \(\boxed{a}\).
Explanation
- Grouping Terms and Identifying the Equation
The goal is to determine the unknown terms \(B_1\) and \(B_2\) that satisfy the following equation:
\[3x^2 + 5y^2 + B_1 + 3 + B_2 + 4y^2 + 6 = 9x^2 - y^2 + 9\]
First, let's group the terms on the left side of the equation by their variables \(x^2\), \(y^2\), and the constants.
Grouping the terms, we get:
\[(3x^2 + B_1) + (5y^2 + 4y^2 + B_2) + (3 + 6) = 9x^2 - y^2 + 9\]
Simplifying the grouped terms:
\[(3x^2 + B_1) + (9y^2 + B_2) + 9 = 9x^2 - y^2 + 9\]
- Solving for Unknown Terms and Selecting the Correct Option
We now compare the coefficients on both sides of the equation.
For the constant terms:
\[9 = 9\]
This is consistent.
For the \(x^2\) terms:
\[3x^2 + B_1 = 9x^2\]
Solving for \(B_1\):
\[B_1 = 9x^2 - 3x^2 = 6x^2\]
For the \(y^2\) terms:
\[9y^2 + B_2 = -y^2\]
Solving for \(B_2\):
\[B_2 = -y^2 - 9y^2 = -10y^2\]
Thus, the terms that need to go into the boxes are \(+6x^2\) and \(-10y^2\). Comparing this result with the given options, it matches option a.
Examples
When designing a garden, you might need to combine different areas to determine the total space. For instance, if you have rectangular plots of varying sizes, you might find that you need to adjust your initial estimates to match a desired total area. This algebraic approach of grouping like terms helps you precisely adjust and manage components of a complex system to reach your final goal.