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Solving the Equation (a+b)^2x^2 - 4abx - (a-b)^2 = 0 using the Quadratic Formula
In this article, we will solve the quadratic equation (a+b)^2x^2 - 4abx - (a-b)^2 = 0 using the quadratic formula.
Understanding the Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations of the form ax^2 + bx + c = 0, where a, b, and c are constants. The formula is:
x = (-b ± √(b^2 - 4ac)) / 2a
This formula will give us two solutions for the value of x.
Expanding the Given Equation
Before we can apply the quadratic formula, we need to expand the given equation:
(a+b)^2x^2 - 4abx - (a-b)^2 = 0
Using the rules of algebra, we can expand the equation as:
a^2x^2 + 2abx^2 + b^2x^2 - 4abx - a^2 + 2ab - b^2 = 0
Combining like terms, we get:
(a^2 + 2ab + b^2)x^2 - 4abx + (2ab - a^2 - b^2) = 0
Rearranging the Equation
To apply the quadratic formula, we need to rearrange the equation in the standard form ax^2 + bx + c = 0. Let's do that:
x^2(a^2 + 2ab + b^2) - x(4ab) + (2ab - a^2 - b^2) = 0
Now, we can identify the values of a, b, and c:
a = a^2 + 2ab + b^2 b = -4ab c = 2ab - a^2 - b^2
Applying the Quadratic Formula
Now, we can plug these values into the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a x = (4ab ± √((-4ab)^2 - 4(a^2 + 2ab + b^2)(2ab - a^2 - b^2))) / 2(a^2 + 2ab + b^2)
Simplifying the expression, we get:
x = (4ab ± √(16a^2b^2 - 8a^2b^2 + 8ab^3 - 8a^3b - 8a^2b^2 - 16ab^3 + 8ab^2)) / 2(a^2 + 2ab + b^2)
x = (4ab ± √(8a^2b^2 - 8a^3b - 8a^2b^2 + 8ab^2)) / 2(a^2 + 2ab + b^2)
x = (4ab ± √(8ab(a^2 - a^2 - ab))) / 2(a^2 + 2ab + b^2)
x = (4ab ± √(0)) / 2(a^2 + 2ab + b^2)
x = (4ab) / 2(a^2 + 2ab + b^2)
x = 2ab / (a^2 + 2ab + b^2)
Thus, we have obtained the solution to the equation (a+b)^2x^2 - 4abx - (a-b)^2 = 0.
Conclusion
In this article, we have successfully solved the quadratic equation (a+b)^2x^2 - 4abx - (a-b)^2 = 0 using the quadratic formula. We expanded the given equation, rearranged it in the standard form, and applied the quadratic formula to obtain the solution.
