(x-1)^2+(x+3)^2=-2(x+5)(3-x)+40

Jasonbradley

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Jun 17, 2024

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Solving the Equation (x-1)^2 + (x+3)^2 = -2(x+5)(3-x) + 40

This article will walk you through the steps of solving the equation (x-1)^2 + (x+3)^2 = -2(x+5)(3-x) + 40. We'll simplify the equation, find the solutions, and explore the process.

Expanding and Simplifying the Equation

First, we need to expand and simplify both sides of the equation:

• Left side:
  (x-1)^2 + (x+3)^2 = (x^2 - 2x + 1) + (x^2 + 6x + 9) = 2x^2 + 4x + 10

• Right side: -2(x+5)(3-x) + 40 = -2(-x^2 - 2x + 15) + 40 = 2x^2 + 4x + 10

Notice that after simplification, both sides of the equation are identical: 2x^2 + 4x + 10 = 2x^2 + 4x + 10

Analyzing the Solution

The resulting equation tells us that both sides are always equal, regardless of the value of x. This means that the original equation has infinite solutions.

Why does this happen?

The original equation is actually a disguised form of an identity. After expanding and simplifying, we end up with a true statement that is always valid, meaning any value of x will satisfy the equation.

Conclusion

The equation (x-1)^2 + (x+3)^2 = -2(x+5)(3-x) + 40 has infinite solutions because it simplifies to an identity. This type of equation demonstrates the importance of simplification and recognizing identities in algebra.