Question

Solve the following by reducing them to quadratic form:
`sqrt(x^2 - 16) - (x - 4) = sqrt(x^2 - 5x + 4)`.

Answer 1

Given equation
`sqrt(x^2 - 16) - (x - 4) = sqrt(x^2 - 5x + 4)`
⇒ `sqrt((x + 4) (x - 4)) - (x - 4) = sqrt((x - 1) (x - 4)`
⇒ `sqrt(x - 4) [sqrt(x + 4) - sqrt(x - 4) - sqrt(x - 1)] = 0`
⇒ Either, `sqrt(x - 4) = 0`
⇒ x - 4 = 0
⇒ x = 4 ...(By squaring on both sides)
or
`sqrt(x + 4) - sqrt(x - 4) - sqrt(x - 1) = 0`
⇒ `sqrt(x + 4) - sqrt(x - 4) = sqrt(x - 1)`
Squaring both sides we get
`x + 4 + x - 4 - 2sqrt((x + 4) (x - 4)) = x - 1`
⇒ `2x - 2sqrt(x^2 - 16)) = x - 1`
⇒ `-2sqrt(x^2 - 16) = x - 2x -1 = -x -1`
= -(x + 1)
⇒ `2sqrt(x^2 - 16)) = x + 1`
Squaring again, 4(x² - 16) = x² + 2x + 1
⇒ 4x² - 64 - x² - 2x - 1 = 0
⇒ 3x² - 2x - 65 = 0
⇒ 3x² - 15x + 13x - 65 = 0
⇒ 3x(x - 5) + 13(x - 5) = 0
⇒ (x - 5) + (3x + 13)  = 0
⇒ x - 5 = 0 or 3x + 13 = 0
⇒ x = 5 or x = `(-13)/(3)`
x = 5.
Hence, the solutions are 4, 5.