To solve for \( x \) in the equation \( 110_x = 40_5 \), we need to convert both numbers to base 10.

### Step 1: Convert \( 40_5 \) to base 10
The subscript "5" indicates that the number is in base 5. So, we interpret \( 40_5 \) as:
\[
40_5 = 4 \times 5^1 + 0 \times 5^0 = 4 \times 5 = 20_{10}
\]

### Step 2: Convert \( 110_x \) to base 10
The subscript "x" indicates that the number is in base \( x \). So, we interpret \( 110_x \) as:
\[
110_x = 1 \times x^2 + 1 \times x^1 + 0 \times x^0 = x^2 + x
\]

### Step 3: Set the equation and solve for \( x \)
Now, equate the two expressions:
\[
x^2 + x = 20
\]
This is a quadratic equation. Rearranging it:
\[
x^2 + x - 20 = 0
\]

### Step 4: Solve the quadratic equation
We can solve \( x^2 + x - 20 = 0 \) using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( x^2 + x - 20 = 0 \), \( a = 1 \), \( b = 1 \), and \( c = -20 \).

Substituting into the formula:
\[
x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-20)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 80}}{2} = \frac{-1 \pm \sqrt{81}}{2} = \frac{-1 \pm 9}{2}
\]

Thus, the two solutions are:
\[
x = \frac{-1 + 9}{2} = \frac{8}{2} = 4
\]
\[
x = \frac{-1 - 9}{2} = \frac{-10}{2} = -5
\]

### Step 5: Determine the valid solution
Since \( x \) must be a positive base, the valid solution is:
\[
x = 4
\]

Thus, \( x = 4 \).

Do you want any further details or have any questions?

Here are 5 related questions to expand your knowledge:
1. What is the method for converting any number from base \( n \) to base 10?
2. How do you solve quadratic equations using factoring instead of the quadratic formula?
3. What would happen if the base \( x \) is not a positive integer?
4. How do you convert numbers from base 10 to any base \( n \)?
5. How can we solve more complex equations with variables in different bases?

**Tip:** Always verify base conversions by converting back to the original base after solving.

Discover how to solve for x in the equation 110_x = 40_5 through base conversion and solving a quadratic equation. Learn key concepts like base-n to base-10 conversion and the quadratic formula. Suitable for students in Grades 8-10.

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