Question:
In a number system of base \(r\), the equation \(x^2 - 12x + 37 = 0\) has \(x = 8\) as one of its solutions. The value of \(r\) is \(\_\_\_\_\).
In a number system of base \(r\), the equation \(x^2 - 12x + 37 = 0\) has \(x = 8\) as one of its solutions. The value of \(r\) is \(\_\_\_\_\).
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When solving equations in a base \(r\), express the coefficients in terms of \(r\) and carefully simplify. Always verify that the solution satisfies the constraints of the given base.
Updated On: Jan 31, 2025
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Solution and Explanation
Step 1: Represent the coefficients in base \(r\).
The coefficients \(12\) and \(37\) are expressed in base \(r\) as: \[ 12 = 1r + 2, \quad 37 = 3r + 7. \] Step 2: Substitute \(x = 8\) into the given equation.
The equation becomes: \[ 8^2 - 12 \cdot 8 + 37 = 0. \] Substituting the base \(r\) expressions for \(12\) and \(37\): \[ 8^2 - (1r + 2) \cdot 8 + (3r + 7) = 0. \] Step 3: Simplify the equation.
Expand and combine terms: \[ 64 - 8r - 16 + 3r + 7 = 0. \] Simplify further: \[ 64 - 16 + 7 = 55 \quad {and} \quad -8r + 3r = -5r. \] Thus: \[ 55 - 5r = 0. \] Step 4: Solve for \(r\).
Rearrange the equation: \[ 5r = 55 \implies r = \frac{55}{5} = 11. \] Final Answer: \[\boxed{{11}}\]
The coefficients \(12\) and \(37\) are expressed in base \(r\) as: \[ 12 = 1r + 2, \quad 37 = 3r + 7. \] Step 2: Substitute \(x = 8\) into the given equation.
The equation becomes: \[ 8^2 - 12 \cdot 8 + 37 = 0. \] Substituting the base \(r\) expressions for \(12\) and \(37\): \[ 8^2 - (1r + 2) \cdot 8 + (3r + 7) = 0. \] Step 3: Simplify the equation.
Expand and combine terms: \[ 64 - 8r - 16 + 3r + 7 = 0. \] Simplify further: \[ 64 - 16 + 7 = 55 \quad {and} \quad -8r + 3r = -5r. \] Thus: \[ 55 - 5r = 0. \] Step 4: Solve for \(r\).
Rearrange the equation: \[ 5r = 55 \implies r = \frac{55}{5} = 11. \] Final Answer: \[\boxed{{11}}\]
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