Find the roots of the equation
2x−31+x−51=1 x =23,5$
(rational equation, roots, domain, excluded values)
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 (rational equation, roots, domain, excluded values)". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Combine fractions
To solve this rational equation, we first combine the fractions on the left side by finding a common denominator. The common denominator for 2x−31 and x−51 is (2x-3)(x-5). We then add the numerators after adjusting them for the common denominator.
Step 2: Simplify and cross-multiply
Next, we simplify the numerator on the left side. Then, we multiply both sides of the equation by the common denominator (2x-3)(x-5) to eliminate the fractions, which is also known as cross-multiplication.
Step 3: Expand and rearrange into standard quadratic form
We expand the right side of the equation by multiplying the terms. After expanding, we combine like terms and rearrange the equation into the standard quadratic form ax2+bx+c=0.
Step 4: 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 5: Substitute values and solve for x
Substitute the values of a, b, and c into the quadratic formula and simplify. We calculate the discriminant, 72, which simplifies to 62. Finally, we divide all terms by 2 to get the simplified roots.
Step 6: Check for excluded values
The problem states that x=23 (which is 1.5) and x=5. Both calculated roots, approximately 6.12 and 1.88, are not equal to the excluded values. Therefore, both are valid roots.