(a+b)^2x^2-4abx-(a-b)^2=0

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Jul 03, 2024

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(a+b)^2x^2-4abx-(a-b)^2=0: Solving the Quadratic Equation

In this article, we will explore the quadratic equation (a+b)^2x^2-4abx-(a-b)^2=0 and provide a step-by-step guide on how to solve it.

Understanding the Equation

The given equation is a quadratic equation in the form of ax^2 + bx + c = 0, where:

• a = (a+b)^2
• b = -4ab
• c = -(a-b)^2

To solve this equation, we can use various methods, including factoring, the quadratic formula, and completing the square. In this article, we will use the factoring method to solve the equation.

Factoring the Equation

Let's start by factoring the left-hand side of the equation:

(a+b)^2x^2 - 4abx - (a-b)^2 = 0

We can factor the first term as:

((a+b)x)^2 - 4abx - (a-b)^2 = 0

Now, let's try to factor the middle term:

-4abx = -2(2ab)x

So, the equation becomes:

((a+b)x)^2 - 2(2ab)x - (a-b)^2 = 0

Solving the Equation

Now, we can factor the equation as:

((a+b)x - (a-b))^2 = 0

This tells us that either:

((a+b)x - (a-b)) = 0 ... (1)

or

((a+b)x - (a-b)) = 0 ... (2)

Solving equation (1), we get:

(a+b)x - (a-b) = 0 (a+b)x = a-b x = (a-b)/(a+b)

And solving equation (2), we get:

-(a+b)x + (a-b) = 0 (a+b)x = -(a-b) x = -(a-b)/(a+b)

Therefore, the solutions to the equation are:

x = (a-b)/(a+b) x = -(a-b)/(a+b)

Conclusion

In this article, we have successfully solved the quadratic equation (a+b)^2x^2-4abx-(a-b)^2=0 using the factoring method. The solutions to the equation are x = (a-b)/(a+b) and x = -(a-b)/(a+b).