k - x = 58 - x In the given equation, k is a constant. The equation has exactly one

Question

k - x = 58 - x In the given equation, k is a constant. The equation has exactly one real solution. What is the minimum possible value of 4 k ?

Accepted Answer

The correct answer is 231 . It's given that k - x = 58 - x . Squaring both sides of this equation yields k - x = ( 58 - x ) 2 , which is equivalent to the given equation if 58 - x &#62; 0 . It follows that if a solution to the equation k - x = ( 58 - x ) 2 satisfies 58 - x &#62; 0 , then it's also a solution to the given equation; if not, it's extraneous. The equation k - x = ( 58 - x ) 2 can be rewritten as k - x = 3,364 - 116 x + x 2 . Adding x to both sides of this equation yields k = x 2 - 115 x + 3,364 . Subtracting k from both sides of this equation yields 0 = x 2 - 115 x + ( 3,364 - k ) . The number of solutions to a quadratic equation in the form 0 = a x 2 + b x + c , where a , b , and c are constants, can be determined by the value of the discriminant, b 2 - 4 a c . Substituting - 115 for b , 1 for a , and 3,364 - k for c in b 2 - 4 a c yields ( - 115 ) 2 - 4 ( 1 ) ( 3,364 - k ) , or 4 k - 231 . The equation 0 = x 2 - 115 x + ( 3,364 - k ) has exactly one real solution if the discriminant is equal to zero, or 4 k - 231 = 0 . Subtracting 231 from both sides of this equation yields 4 k = 231 . Dividing both sides of this equation by 4 yields k = 57.75 . Therefore, if k = 57.75 , then the equation 0 = x 2 - 115 x + ( 3,364 - k ) has exactly one real solution. Substituting 57.75 for k in this equation yields 0 = x 2 - 115 x + ( 3,364 - 57.75 ) , or 0 = x 2 - 115 x + 3,306.25 , which is equivalent to 0 = ( x - 57.5 ) 2 . Taking the square root of both sides of this equation yields 0 = x - 57.5 . Adding 57.5 to both sides of this equation yields 57.5 = x . To check whether this solution satisfies 58 - x &#62; 0 , the solution, 57.5 , can be substituted for x in 58 - x &#62; 0 , which yields 58 - 57.5 &#62; 0 , or 0.5 &#62; 0 . Since 0.5 is greater than 0 , it follows that if k = 57.75 , or 4 k = 231 , then the given equation has exactly one real solution. If 4 k &#60; 231 , then the discriminant, 4 k - 231 , is negative and the given equation has no solutions. Therefore, the minimum possible value of 4 k is 231 .

Question k-x-58