Simplifying Radicals and Solving for x
The problem requires simplifying the expression $\sqrt{80} + 4\sqrt{5}$ using the given equation $2\sqrt{5} + \sqrt{125} = 7x$. We need to find the value of $x$ first.
Step 1: Simplify the given equation
First, simplify the terms in the equation $2\sqrt{5} + \sqrt{125} = 7x$.
- Calculate $\sqrt{125}$: $\sqrt{125} = \sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5} = 5\sqrt{5}$.
- Substitute this back into the equation: $2\sqrt{5} + 5\sqrt{5} = 7x$.
- Combine the terms: $7\sqrt{5} = 7x$.
- Solve for $x$: Divide both sides by 7, which gives $x = \sqrt{5}$.
Step 2: Simplify the expression to be calculated
Next, simplify the expression $\sqrt{80} + 4\sqrt{5}$.
- Calculate $\sqrt{80}$: $\sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5}$.
- Substitute this simplified term back into the expression: $4\sqrt{5} + 4\sqrt{5}$.
- Combine the terms: $8\sqrt{5}$.
Step 3: Express the result in terms of x
We found the expression equals $8\sqrt{5}$. We know from Step 1 that $x = \sqrt{5}$.
- Substitute $x$ for $\sqrt{5}$ in the result: $8\sqrt{5} = 8x$.
Therefore, the value of $\sqrt{80} + 4\sqrt{5}$ is $8x$. This matches Option C.
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