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

Question

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

(2x+a)(a4)=2ax+a25 (2x+a)(a-4)=2ax+a^2-5

Video Solution

Step-by-Step Solution

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 a and x x .

Now, let's work through each step:

Step 1:
The given equation is (2x+a)(a4)=2ax+a25 (2x+a)(a-4) = 2ax + a^2 - 5 . First, expand the left-hand side:

(2x+a)(a4) (2x + a)(a - 4)

Using the distributive property:

  • 2xa=2ax 2x \cdot a = 2ax
  • 2x4=8x 2x \cdot -4 = -8x
  • aa=a2 a \cdot a = a^2
  • a4=4a a \cdot -4 = -4a

Combining these terms gives:

2ax8x+a24a 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:

2ax8x+a24a=2ax+a25 2ax - 8x + a^2 - 4a = 2ax + a^2 - 5

Cancel the common terms on both sides:

  • Subtract 2ax 2ax from both sides: 8x4a=5 -8x - 4a = -5
  • Subtract a2 a^2 from both sides: No change needed since both sides already equal.

The equation becomes:

8x4a=5 -8x - 4a = -5

Solving for a a :

Add 4a 4a to both sides:

8x=4a5 -8x = 4a - 5

Divide each term by 4 to solve for a a :

a=2x+54 a = -2x + \frac{5}{4}

Expressing in a simpler equivalent format, we have:

a=2x+114 a = -2x + 1\frac{1}{4}

Therefore, we find the relationship between a a and x x to be a=1142x a = 1\frac{1}{4} - 2x .

Answer

a=1142x a=1\frac{1}{4}-2x