Quadratic Equations-29

From Homeworkwiki

Solve the equation: √(5x² - 6x + 8) - √( 5x² - 6x – 7) = 1

Solution: Let 5x² - 6x = y. Then,

√(5x² - 6x + 8) - √( 5x² - 6x – 7) = 1

=>√(y + 8) - √(y – 7) = 1

=> [√(y + 8) - √(y – 7)]² = 1

=> y + 8 + y – 7 – 2√(y² + y – 56) = 1

=> 2y + 1 = -2√(y² + y – 56) + 1

=> y = -√(y² + y – 56)

=> y² = y² + y – 56

=> y = 56

=> 5x² - 6x = 56

=> 5x² - 6x – 56 = 0

=> (5x + 14)(x - 4) = 0

=> x = 4, -14/5

Clearly, both the values of x satisfy the given equation. Hence, the roots of the given equation are 4, -14/5.