Tricky algebra problem
Solve for x in the equation:
(2x + 3)/(x – 2) = (3x + 4)/(x + 1)
Word Problem
A bridge is built across a river. One quarter of the bridge is over the left bank of the river and one third of the bridge is over the right bank. The river is 120 m wide. How long is the bridge?
Step 1: Cross Multiply
ReplyDeleteCross multiply to eliminate the fractions:
(2x + 3)(x + 1) = (3x + 4)(x - 2)
Step 2: Expand Both Sides
Expanding the left-hand side:
2x * x + 2x * 1 + 3 * x + 3 * 1
2x² + 2x + 3x + 3 = 2x² + 5x + 3
Expanding the right-hand side:
3x * x + 3x * (-2) + 4 * x + 4 * (-2)
3x² - 6x + 4x - 8 = 3x² - 2x - 8
Step 3: Set the Equation to Zero
2x² + 5x + 3 = 3x² - 2x - 8
Subtract everything on the right-hand side from both sides:
(2x² + 5x + 3) - (3x² - 2x - 8) = 0
2x² + 5x + 3 - 3x² + 2x + 8 = 0
-x² + 7x + 11 = 0
Multiply by -1 to simplify:
x² - 7x - 11 = 0
Step 4: Solve Using the Quadratic Formula
For the equation:
x² - 7x - 11 = 0
Use the quadratic formula:
x = (-(-7) ± √((-7)² - 4(1)(-11))) / (2(1))
x = (7 ± √(49 + 44)) / 2
x = (7 ± √93) / 2
Since √93 ≈ 9.64, we get:
x = (7 ± 9.64) / 2
Step 5: Find the Two Possible Values for x
x = (7 + 9.64) / 2 = 16.64 / 2 = 8.32
x = (7 - 9.64) / 2 = -2.64 / 2 = -1.32
Final Answer
x = 8.32 or x = -1.32