Solve the quadratic equation 6x^{2} - x - 1 = 0 using the formula method

Solution

The standard form of the quadratic equation is ax^{2} + bx + c = 0. Compare the given equation with the standard equation, we get, a = 6, b = -1 and c = -1.

The formula to solve quadratic equation is x_{1,2} = [-b + - square root of b^{2} - 4ac]/2a

substitude a, b and c into the formula

x_{1,2} = [-b + - square root of b^{2} - 4ac]/2a

= [-(-1) + - square root of (-1)^{2} - 4(6)(-1)]/2(6)

= [1 + - square root of 1 + 24]/12

= [1 + - square root of 25/12

= [1 + - 5]/12

so, we get two equations, one is x = (1 + 5)/12 and other is (1 - 5)/12

if x = (1 + 5)/12, then x = 6/12 = 1/2

if x = (1 - 5)/12, then x = -4/12 = -1/3

so, we get two solutions, x_{1} = 1/2 and x_{2} = -1/3

Substitute x_{1} = 1/2 and x_{2} = -1/3 into the original equation, find that the left side of the equation is equal to the right side of the equation.
So, x_{1} = 1/2 and x_{2} = -1/3 are the solutions of original equation.