A substitution helps when the same expression appears unchanged on both sides of an equation and is also what the question asks for. Give the whole expression one name, then solve the shorter equation with a Desmos regression. Don’t distribute and solve for the inner variable when you don’t need it, and don’t stop at a multiple of the requested expression.
Hints
- Hint 1
The parentheses appear unchanged on both sides, so treat everything inside them as one number. What shorter equation do you get if you replace each copy of the parentheses with ?
- Hint 2
A regression uses in place of and reports the unknown’s value under PARAMETERS. Here stands for the whole parenthesis. What value does Desmos find for it?
Step-by-step
Solve for the repeated expression
Step 1Give the repeated expression one name
Call the whole expression :
It appears unchanged on both sides, so replace each copy with :
This substitution avoids finding , which the question doesn’t ask for.
- Step 2
Find the value of the whole expression
Change to and type . This tells Desmos to find the that makes both sides equal. Under PARAMETERS, it shows . Since stands for the whole expression , the requested value is . Choice C.