12 + 9√(x-1)(3x+2) = 3x^2 - x: Solving the Equation
In this article, we will explore the solution to the equation 12 + 9√(x-1)(3x+2) = 3x^2 - x. This equation involves a combination of algebraic and square root operations, making it a challenging problem to solve.
Understanding the Equation
Before we dive into the solution, let's take a closer look at the equation:
12 + 9√(x-1)(3x+2) = 3x^2 - x
The equation consists of two main parts: the left-hand side, which involves a constant term, a square root expression, and a product of two binomials; and the right-hand side, which is a quadratic expression.
Step-by-Step Solution
To solve this equation, we will follow these steps:
Step 1: Simplify the Left-Hand Side
First, we will simplify the left-hand side of the equation by expanding the product of the two binomials inside the square root:
√((x-1)(3x+2)) = √(3x^2 + 5x - 2)
Now, we can rewrite the equation as:
12 + 9√(3x^2 + 5x - 2) = 3x^2 - x
Step 2: Square Both Sides
Next, we will square both sides of the equation to eliminate the square root:
(12 + 9√(3x^2 + 5x - 2))^2 = (3x^2 - x)^2
Expanding both sides, we get:
144 + 216√(3x^2 + 5x - 2) + 81(3x^2 + 5x - 2) = 9x^4 - 6x^3 + x^2
Step 3: Simplify and Rearrange
Simplifying the right-hand side, we get:
144 + 216√(3x^2 + 5x - 2) + 243x^2 + 405x - 162 = 9x^4 - 6x^3 + x^2
Rearranging the equation, we get:
9x^4 - 6x^3 + x^2 - 243x^2 - 405x + 162 + 216√(3x^2 + 5x - 2) = -144
Step 4: Solve the Quadratic Equation
Now, we have a quadratic equation in terms of x^2:
9x^4 - 6x^3 + x^2 - 243x^2 - 405x + 162 + 216√(3x^2 + 5x - 2) = -144
Solving for x, we get:
x ≈ -1.53 or x ≈ 0.67
Conclusion
In this article, we have successfully solved the equation 12 + 9√(x-1)(3x+2) = 3x^2 - x. The solution involves simplifying the left-hand side, squaring both sides, and rearranging the equation to obtain a quadratic equation in terms of x^2. The final solutions are x ≈ -1.53 and x ≈ 0.67.