Theory question 110
Surds(a) Solve the equation √(4x − 3) − √(2x − 5) = 2
(b) Find the finite area enclosed by the curve y2 = 4x and the line y + x = 0.
Show the model answer
Model answer
(a) √(4x − 3) = 2 + √(2x − 5); square: 4x − 3 = 4 + 4√(2x − 5) + 2x − 5
2x − 2 = 4√(2x − 5), so x − 1 = 2√(2x − 5)
Square: x² − 2x + 1 = 8x − 20, x² − 10x + 21 = 0, x = 3 or x = 7
Check: x = 3 gives 3 − 1 = 2; x = 7 gives 5 − 3 = 2; both are solutions
(b) Intersections: x = −y, y² = −4y, so y = 0 or y = −4; points (0, 0) and (4, −4)
Area = ∫−40 (−y − y²/4) dy = [−y²/2 − y³/12] from −4 to 0 = 0 − (−8 + 16/3) = 8/3 square units
Explanation
Square twice and check both answers in the original equation; for the area integrate with respect to y.