Compute the sum of all the roots of (2x+3)(x-4)+(2x+3)(x-6)=0
Hint 1: Do not expand the quadratic terms immediately. Notice the common factor .
Hint 2: Factor out and simplify the remaining expression inside the brackets.
Hint 3: Find the two roots of the resulting linear factors and add them together.
Step 1 (Factor out Common Term): We are given the equation:
Notice that is a common binomial factor in both terms. We factor it out:
Step 2 (Simplify the Equation): Combine the terms inside the bracket:
Dividing the second term by 2:
Step 3 (Find the Roots): Set each factor to zero:
Step 4 (Calculate Sum of Roots):
Step 5 (Conclusion): The sum of all the roots is exactly 3.5 (or 7/2).
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