Solve (2x+a)(a-4)=2ax+a²-5: Extended Distributive Law Application

Solve the equation using the extended distributive law. Find the relationship between a and x.

$(2x+a)(a-4)=2ax+a^2-5$

To solve this problem, we'll follow these steps:

• Step 1: Expand the expression using the distributive property.
• Step 2: Compare the coefficients and constant terms to solve for a relationship between $a$ and $x$.

Now, let's work through each step:

Step 1:
The given equation is $(2x+a)(a-4) = 2ax + a^2 - 5$. First, expand the left-hand side:

$(2x + a)(a - 4)$

Using the distributive property:

• $2x \cdot a = 2ax$
• $2x \cdot -4 = -8x$
• $a \cdot a = a^2$
• $a \cdot -4 = -4a$

Combining these terms gives:

$2ax - 8x + a^2 - 4a$

Step 2:
Now, we set the expanded left-hand side equal to the right-hand side from the original equation:

$2ax - 8x + a^2 - 4a = 2ax + a^2 - 5$

Cancel the common terms on both sides:

• Subtract $2ax$ from both sides: $-8x - 4a = -5$
• Subtract $a^2$ from both sides: No change needed since both sides already equal.

The equation becomes:

$-8x - 4a = -5$

Solving for $a$:

Add $4a$ to both sides:

$-8x = 4a - 5$

Divide each term by 4 to solve for $a$:

$a = -2x + \frac{5}{4}$

Expressing in a simpler equivalent format, we have:

$a = -2x + 1\frac{1}{4}$

Therefore, we find the relationship between $a$ and $x$ to be $a = 1\frac{1}{4} - 2x$.