Solve the equation using the introduction of a new variable (6x-1) ^ 2-7 (6x-1) -144 = 0

To find a solution to the equation (6x – 1) ^ 2 – 7 (6x – 1) – 144 = 0, we start by introducing a variable.

We denote by the variable t = 6x – 1 and we get the following equation:

t ^ 2 – 7t – 144 = 0;

We solve the resulting full quadratic equation through the discriminant:

D = b ^ 2 – 4ac = (-7) ^ 2 – 4 * 1 * (-144) = 49 + 576 = 625;

We calculate the roots of the equation using the following formulas:

t1 = (-b + √D) / 2a = (7 + √625) / 2 * 1 = (7 + 25) / 2 = 32/2 = 16;

t2 = (-b – √D) / 2a = (7 – 25) / 2 * 1 = (7 – 25) / 2 = 18/2 = 9.

Let’s go back to the replacement:

1) 6x – 1 = 16;

6x = 16 + 1;

6x = 17;

x = 17: 6;

x = 2 5/6;

2) 6x – 1 = 9;

6x = 10;

x = 10: 6;

x = 1 2/3.