Find the roots of the equation
2x−31+x−51=1
x=23,5
(roots, equation, fractions, domain, quadratic)
Get the complete, step-by-step math solution for: "Find the roots of the equation (1)/(2x-3) + (1)/(x-5) = 1 x ≠ (3)/(2), 5 (roots, equation, fractions, domain, quadratic)". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Combine fractions on the left side
To solve this equation, we first need to combine the fractions on the left-hand side. We do this by finding a common denominator, which is the product of the two denominators: (2x-3)(x-5). Then, we cross-multiply the numerators with the opposite denominators and add them.
Step 2: Simplify the numerator and cross-multiply
Now, we simplify the numerator on the left side by combining like terms. Then, we multiply both sides of the equation by the denominator (2x-3)(x-5) to eliminate the fraction, moving it to the right side.
Step 3: Expand the right side
We expand the product of the two binomials on the right side of the equation using the distributive property (FOIL method). This will result in a quadratic expression.
Step 4: Rearrange into standard quadratic form
To solve the quadratic equation, we need to rearrange it into the standard form ax2+bx+c=0. We do this by moving all terms to one side of the equation and combining like terms.
Step 5: Apply the quadratic formula
Since the quadratic equation 2x2−16x+23=0 cannot be easily factored, we use the quadratic formula to find the roots. Here, a=2, b=−16, and c=23.
Step 6: Substitute values and simplify
Substitute the values of a, b, and c into the quadratic formula and simplify the expression. Remember to simplify the square root and then divide all terms by the common factor in the numerator and denominator.