Solving the Equation ( sqrt{x-1} x-m ) Using Quadratic Formulas and Methods
" "This guide will walk you through the process of solving the equation ( sqrt{x-1} x - m ) using both algebraic manipulation and the quadratic formula. We'll start by squaring both sides to eliminate the square root, then derive a quadratic equation in terms of ( x ) and solve for ( m ).
" "Step 1: Square Both Sides
" "To eliminate the square root, we can square both sides of the equation:
" "[ sqrt{x-1} x - m ]
" "Squaring both sides:
" "[ (sqrt{x-1})^2 (x - m)^2 ]
" "Simplifies to:
" "[ x - 1 x^2 - 2mx m^2 ]
" "Rearrange the equation to standard quadratic form:
" "[ x^2 - 2mx - x m^2 1 0 ]
" "Simplify further:
" "[ x^2 - (2m 1)x (m^2 1) 0 ]
" "Step 2: Solve the Quadratic Equation
" "Now we can solve the quadratic equation using the quadratic formula:
" "[ x frac{-b pm sqrt{b^2 - 4ac}}{2a} ]
" "In this equation, ( a 1 ), ( b -(2m 1) ), and ( c (m^2 1) ).
" "Substitute these values into the quadratic formula:
" "[ x frac{-( -(2m 1)) pm sqrt{(-(2m 1))^2 - 4(1)(m^2 1)}}{2(1)} ]
" "Simplify the expression:
" "[ x frac{(2m 1) pm sqrt{(2m 1)^2 - 4(m^2 1)}}{2} ]
" "Further manipulation gives:
" "[ x frac{2m 1 pm sqrt{4m^2 4m 1 - 4m^2 - 4}}{2} ]
" "[ x frac{2m 1 pm sqrt{4m - 3}}{2} ]
" "Therefore, the solutions for ( x ) are:
" "[ x frac{2m 1 sqrt{4m - 3}}{2} ]
" "[ x frac{2m 1 - sqrt{4m - 3}}{2} ]
" "Note that not all solutions may be valid, as some could result in extraneous roots. You need to check the valid solutions by plugging them back into the original equation.
" "Step 3: Solve for ( m )
" "To find ( m ) in terms of ( x ), simply rearrange the original equation:
" "[ sqrt{x-1} x - m ]
" "Add ( sqrt{x-1} ) to both sides:
" "[ m x - sqrt{x-1} ]
" "Therefore, ( m ) is determined by the following relationship:
" "[ m x - sqrt{x-1} ]
" "Conclusion
" "This guide has provided a comprehensive solution to the given equation using both algebraic manipulations and the quadratic formula. The key steps include squaring both sides, forming a quadratic equation, and solving for ( x ) and ( m ).
" "For further exploration, consider testing the derived solutions for valid ( x ) values and understanding the conditions under which ( m ) is defined.