If (3a + 6b + c + 2d) × (3a - 6b - c + 2d) = (3a - 6b + c - 2d) × (3a + 6b - c - 2d), then which one of the following is correct?
ad = bc
Solving the Algebraic Equation
The problem asks us to find the relationship between the variables a, b, c, and d given a specific equation involving these terms. The equation is:
\[(3a + 6b + c + 2d) \times (3a - 6b - c + 2d) = (3a - 6b + c - 2d) \times (3a + 6b - c - 2d)\]
We can solve this by recognizing the pattern of the difference of squares formula: \[(x+y)(x-y) = x^2 - y^2\]
Applying Difference of Squares to the Left Side
Consider the left side of the equation: \[(3a + 6b + c + 2d) \times (3a - 6b - c + 2d)\]
We can group terms to match the \((x+y)(x-y)\) pattern.
Let \(x_L = 3a + 2d\) and \(y_L = 6b + c\).
Then the left side is \[(x_L + y_L)(x_L - y_L) = x_L^2 - y_L^2\]
Substituting back the expressions for \(x_L\) and \(y_L\):
\[x_L^2 - y_L^2 = (3a + 2d)^2 - (6b + c)^2\]
Expand the terms using the formula \((p+q)^2 = p^2 + 2pq + q^2\):
\[(3a + 2d)^2 = (3a)^2 + 2(3a)(2d) + (2d)^2 = 9a^2 + 12ad + 4d^2\]
\[(6b + c)^2 = (6b)^2 + 2(6b)(c) + c^2 = 36b^2 + 12bc + c^2\]
So, the left side simplifies to:
\[(9a^2 + 12ad + 4d^2) - (36b^2 + 12bc + c^2)\]
\[= 9a^2 + 12ad + 4d^2 - 36b^2 - 12bc - c^2\]
Applying Difference of Squares to the Right Side
Now consider the right side of the equation: \[(3a - 6b + c - 2d) \times (3a + 6b - c - 2d)\]
Again, group terms to match the \((x-y)(x+y)\) pattern.
Let \(x_R = 3a - 2d\) and \(y_R = 6b - c\).
Then the right side is \[(x_R + y_R)(x_R - y_R) = x_R^2 - y_R^2\]
Substituting back the expressions for \(x_R\) and \(y_R\):
\[x_R^2 - y_R^2 = (3a - 2d)^2 - (6b - c)^2\]
Expand the terms using the formula \((p-q)^2 = p^2 - 2pq + q^2\):
\[(3a - 2d)^2 = (3a)^2 - 2(3a)(2d) + (2d)^2 = 9a^2 - 12ad + 4d^2\]
\[(6b - c)^2 = (6b)^2 - 2(6b)(c) + c^2 = 36b^2 - 12bc + c^2\]
So, the right side simplifies to:
\[(9a^2 - 12ad + 4d^2) - (36b^2 - 12bc + c^2)\]
\[= 9a^2 - 12ad + 4d^2 - 36b^2 + 12bc - c^2\]
Equating the Left and Right Sides
Now, we set the simplified left side equal to the simplified right side:
\[9a^2 + 12ad + 4d^2 - 36b^2 - 12bc - c^2 = 9a^2 - 12ad + 4d^2 - 36b^2 + 12bc - c^2\]
We can cancel out the terms that appear on both sides of the equation. The terms \(9a^2\), \(4d^2\), \(-36b^2\), and \(-c^2\) are present on both sides.
After cancelling these terms, the equation becomes:
\[12ad - 12bc = -12ad + 12bc\]
Now, let's rearrange the terms to gather the 'ad' terms on one side and the 'bc' terms on the other side. Add \(12ad\) to both sides and add \(12bc\) to both sides:
\[12ad + 12ad = 12bc + 12bc\]
\[24ad = 24bc\]
Divide both sides by 24:
\[ad = bc\]
Comparing with Options
The relationship derived from the equation is \(ad = bc\). Let's check the given options:
- Option 1: \(ab = cd\)
- Option 2: \(ac = bd\)
- Option 3: \(ad = bc\)
- Option 4: \(ad + bc = 0\)
Our result, \(ad = bc\), matches Option 3.
Revision Table: Key Concepts
| Concept | Description | Formula |
|---|---|---|
| Difference of Squares | A fundamental algebraic identity used to factor or simplify expressions. | \((x+y)(x-y) = x^2 - y^2\) |
| Expanding Squares | Multiplying a binomial by itself. | \((p+q)^2 = p^2 + 2pq + q^2\) \((p-q)^2 = p^2 - 2pq + q^2\) |
| Solving Equations | Finding the values or relationships between variables that make an equation true by isolating terms or simplifying both sides. | Maintain balance by performing the same operation on both sides. |
Additional Information: Properties Used in Solving Equations
When solving algebraic equations, we use several properties of equality to manipulate the equation while keeping it balanced. Some properties used in this solution include:
- Addition Property of Equality: If \(A = B\), then \(A + C = B + C\). We added \(12ad\) and \(12bc\) to both sides.
- Subtraction Property of Equality: If \(A = B\), then \(A - C = B - C\). We used this implicitly when cancelling terms that appeared on both sides (e.g., subtracting \(9a^2\) from both sides).
- Division Property of Equality: If \(A = B\) and \(C \neq 0\), then \(\frac{A}{C} = \frac{B}{C}\). We divided both sides by 24.
Understanding these properties is crucial for correctly manipulating algebraic expressions and solving equations.
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