(2x+1)(3x+2)=6(x-1)(x-2) Is A Quadratic Equation

Jasonbradley

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Jun 16, 2024

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Understanding Quadratic Equations: A Case Study

A quadratic equation is an equation that can be written in the standard form: ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. This equation has a maximum of two solutions, also known as roots. Let's explore why the equation (2x+1)(3x+2) = 6(x-1)(x-2) is indeed a quadratic equation.

Expanding the Equation

To understand why the given equation is a quadratic, we need to expand and simplify it.

1. Expand the products:

   • (2x+1)(3x+2) = 6x² + 7x + 2
   • 6(x-1)(x-2) = 6(x² - 3x + 2) = 6x² - 18x + 12

2. Substitute the expanded expressions back into the original equation: 6x² + 7x + 2 = 6x² - 18x + 12

3. Simplify by subtracting 6x² from both sides: 7x + 2 = -18x + 12

4. Combine like terms by adding 18x to both sides: 25x + 2 = 12

5. Isolate the x term by subtracting 2 from both sides: 25x = 10

6. Solve for x by dividing both sides by 25: x = 2/5

This demonstrates that the equation can be rewritten in the standard form of a quadratic equation.

Key Points

• The equation (2x+1)(3x+2) = 6(x-1)(x-2) is a quadratic equation because after simplification, it can be written in the form ax² + bx + c = 0, where a ≠ 0.
• The expansion process helps us understand the relationship between the original equation and its simplified quadratic form.
• The simplified quadratic equation 25x - 10 = 0 has one solution, which is x = 2/5.

While the initial form of the equation might seem complex, understanding the process of simplification and identifying the quadratic form allows us to use the tools of quadratic equations to solve for the unknown variable, x.