12+9√(x-1)(3x+2)=3x^2-x

4 min read Jun 26, 2024

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.