Solve an equation example

Question:

Solve the equation: 12 x + square root of x - 1 = 0

Solution:

12 x + square root of x - 1 = 0

note: x = (square root of x)²

12 (square root of x)² + (square root of x) - 1 = 0

let t = (square root of x)

12 t² + t - 1 = 0

using formula, the roots of ax² + bx + c = 0 is: x_1,2 = [-b +- square root of (b² - 4ac)]/2a. From the one variable second degree equation 12 t² + t - 1 = 0, we get, a = 12, b = 1 c = -1

t_1,2 = [-b +- square root of (b² - 4ac)]/2a

= [-1 +- square root of (1² - 4 × 12 × (-1))]/2 × 12

= [-1 +- square root of (1 + 48)]/24

= [-1 +- square root of 49]/24

= [-1 +- 7]/24

t₁ = (-1 + 7)/24 = 6/24 = 1/4

square root of x = 1/4

x = (1/4)² = 1/16

t₂ = (-1 - 7)/24 = -8/24 = -1/3

square root of x cannot be a negative number, so drop the negative solution.

Substitute x = 1/16 into the given equation, find the left side of the equation is equal to the right side of the equation, so x = 1/16 is the root of given equation.