Question

∫ √(2x² - 5x + 2) dx = ∫ (41/60) dx,

and

-1/2 > α > 0, then α = ?

• A. \(-\tfrac{5}{31}\)
• B. \(-\tfrac{7}{34}\)
• C. \(-\tfrac{9}{37}\)
• D. \(-\tfrac{11}{41}\)

Hint:

Always check for extraneous solutions when squaring equations involving radicals.
Use the interval constraint \(-\tfrac12<x<0\) to pick the correct root

Answer 1

The Correct Option is B

Step 1: Eliminate the radicals and rearrange.

Given the equation:

\[ \frac{x - 1}{\sqrt{2x^2 - 5x + 2}} = \frac{41}{60}, \]

cross-multiply to obtain:

\[ 60(x - 1) = 41\sqrt{2x^2 - 5x + 2}. \]

Next, square both sides (being careful) and move all terms to one side to form a polynomial equation in \(x\).

Step 2: Solve the resulting equation.

Expanding both sides, we get:

\[ 3600(x - 1)^2 = 1681(2x^2 - 5x + 2). \]

Simplify this expression and solve for \(x\). This process should yield two real solutions, though there may be extraneous solutions to check.

Step 3: Select the root in the interval \(-\tfrac{1}{2} > x > 0\).

Among the real solutions, determine which one falls between \(-\tfrac{1}{2}\) and \(0\). The correct root is \(\boxed{-\tfrac{7}{34}}\).