Solve the equation using the extended distributive law. Find the relationship between a and x.
(2x+a)(a−4)=2ax+a2−5
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+a2−5. First, expand the left-hand side:
(2x+a)(a−4)
Using the distributive property:
- 2x⋅a=2ax
- 2x⋅−4=−8x
- a⋅a=a2
- a⋅−4=−4a
Combining these terms gives:
2ax−8x+a2−4a
Step 2:
Now, we set the expanded left-hand side equal to the right-hand side from the original equation:
2ax−8x+a2−4a=2ax+a2−5
Cancel the common terms on both sides:
- Subtract 2ax from both sides: −8x−4a=−5
- Subtract a2 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+45
Expressing in a simpler equivalent format, we have:
a=−2x+141
Therefore, we find the relationship between a and x to be a=141−2x.
a=141−2x