Let's solve the equation step by step:
The equation is:
[
(3x - 1)(3x + 1) - (x - 1)(x + 2) = 8
]
First, simplify the left side of the equation:
Expand ((3x - 1)(3x + 1)):
[
(3x - 1)(3x + 1) = 9x^2 - 1
]
Expand ((x - 1)(x + 2)):
[
(x - 1)(x + 2) = x^2 + 2x - x - 2 = x^2 + x - 2
]
Now substitute these expansions back into the equation:
[
9x^2 - 1 - (x^2 + x - 2) = 8
]
Distribute the negative sign:
[
9x^2 - 1 - x^2 - x + 2 = 8
]
Combine like terms:
[
9x^2 - x^2 - x - 1 + 2 = 8
]
[
8x^2 - x + 1 = 8
]
Now, bring everything to one side of the equation:
[
8x^2 - x + 1 - 8 = 0
]
[
8x^2 - x - 7 = 0
]
Now we can use the quadratic formula to solve for (x):
The quadratic formula is given by:
[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
]
Here, (a = 8), (b = -1), and (c = -7).
First, calculate the discriminant:
[
b^2 - 4ac = (-1)^2 - 4(8)(-7) = 1 + 224 = 225
]
Now substitute this back into the formula:
[
x = \frac{-(-1) \pm \sqrt{225}}{2(8)}
]
[
x = \frac{1 \pm 15}{16}
]
This gives us two potential solutions:
1.
[
x_1 = \frac{1 + 15}{16} = \frac{16}{16} = 1
]
- [
x_2 = \frac{1 - 15}{16} = \frac{-14}{16} = -\frac{7}{8}
]
Thus, the solutions are:
[
x_1 = 1, \quad x_2 = -\frac{7}{8}
]
