To solve the equation \(314_{10} - 256_{7} = 340_{x}\), we need to convert all numbers to the same base, preferably base 10 (decimal), so we can perform the arithmetic operations easily. Let's break this down step-by-step. ### Step 1: Convert \(256_{7}\) to Decimal (Base 10) The number \(256_{7}\) is in base 7. To convert it to decimal, we use the formula for base conversion: \[ n = d_k \cdot b^k + d_{k-1} \cdot b^{k-1} + \ldots + d_1 \cdot b^1 + d_0 \cdot b^0 \] Where \(d\) are the digits of the number, \(b\) is the base, and \(k\) is the position of the digit (starting from 0 on the right). For \(256_{7}\): - The digit \(2\) is in the \(7^2\) place, - The digit \(5\) is in the \(7^1\) place, - The digit \(6\) is in the \(7^0\) place. Calculating each term: \[ 2 \cdot 7^2 = 2 \cdot 49 = 98 \] \[ 5 \cdot 7^1 = 5 \cdot 7 = 35 \] \[ 6 \cdot 7^0 = 6 \cdot 1 = 6 \] Now, we sum these values: \[ 256_{7} = 98 + 35 + 6 = 139_{10} \] ### Step 2: Perform the Subtraction in Decimal Now we can substitute back into our original equation: \[ 314_{10} - 139_{10} = 340_{x} \] Calculating the left side: \[ 314 - 139 = 175 \] ### Step 3: Convert \(340_{x}\) to Decimal Next, we need to express \(340_{x}\) in decimal. Using the same base conversion formula: For \(340_{x}\): - The digit \(3\) is in the \(x^2\) place, - The digit \(4\) is in the \(x^1\) place, - The digit \(0\) is in the \(x^0\) place. Thus, we have: \[ 340_{x} = 3 \cdot x^2 + 4 \cdot x^1 + 0 \cdot x^0 = 3x^2 + 4x \] ### Step 4: Set Up the Equation Now we set the two sides equal to each other: \[ 175 = 3x^2 + 4x \] ### Step 5: Rearranging the Equation Rearranging gives us a standard quadratic equation: \[ 3x^2 + 4x - 175 = 0 \] ### Step 6: Solve the Quadratic Equation We can use the quadratic formula to solve for \(x\): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Where \(a = 3\), \(b = 4\), and \(c = -175\). Calculating the discriminant: \[ b^2 - 4ac = 4^2 - 4 \cdot 3 \cdot (-175) = 16 + 2100 = 2116 \] Now, taking the square root: \[ \sqrt{2116} = 46 \] Now substituting back into the quadratic formula: \[ x = \frac{-4 \pm 46}{2 \cdot 3} = \frac{-4 \pm 46}{6} \] Calculating the two possible values for \(x\): 1. \(x = \frac{42}{6} = 7\) 2. \(x = \frac{-50}{6}\) (which is negative and not valid for a base) Thus, the only valid solution is: \[ x = 7 \] ### Conclusion The correct option is **A. 7**. ### Why Other Options Are Incorrect - **B. 8**: If \(x = 8\), substituting back into \(3x^2 + 4x\) gives \(3(8^2) + 4(8) = 192 + 32 = 224\), which does not equal 175. - **C. 9**: If \(x = 9\), substituting gives \(3(9^2) + 4(9) = 243 + 36 = 279\), which is also incorrect. - **D. 10**: If \(x = 10\), substituting gives \(3(10^2) + 4(10) = 300 + 40 = 340\), which is not equal to 175. ### Revision Summary - Convert numbers from their original bases to decimal for easier calculations. - Use the quadratic formula to solve for unknowns in equations. - Check each option by substituting back into the original equation to verify correctness. - Remember that bases must be positive integers greater than 1.