(x-5)(x+5)-(x+3)^2+3(x-2)^2=(x+1)^2-(x+4)(x-4)+3x^2

Jasonbradley

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Jun 17, 2024

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Solving the Equation: (x-5)(x+5)-(x+3)^2+3(x-2)^2=(x+1)^2-(x+4)(x-4)+3x^2

This article will guide you through the process of solving the equation: (x-5)(x+5)-(x+3)^2+3(x-2)^2=(x+1)^2-(x+4)(x-4)+3x^2

1. Expanding the Equation

Begin by expanding all the brackets using the distributive property or relevant algebraic identities:

• Difference of Squares: (a-b)(a+b) = a^2 - b^2
• Square of a Sum: (a+b)^2 = a^2 + 2ab + b^2

Applying these to our equation:

(x-5)(x+5) - (x+3)^2 + 3(x-2)^2 = (x+1)^2 - (x+4)(x-4) + 3x^2

This expands to:

x^2 - 25 - (x^2 + 6x + 9) + 3(x^2 - 4x + 4) = x^2 + 2x + 1 - (x^2 - 16) + 3x^2

2. Simplifying the Equation

Now, simplify the equation by removing the parentheses and combining like terms:

x^2 - 25 - x^2 - 6x - 9 + 3x^2 - 12x + 12 = x^2 + 2x + 1 - x^2 + 16 + 3x^2

This simplifies to:

3x^2 - 18x - 22 = 3x^2 + 2x + 17

3. Solving for x

Next, move all terms to one side of the equation to set it equal to zero:

3x^2 - 18x - 22 - 3x^2 - 2x - 17 = 0

This simplifies to:

-20x - 39 = 0

Now, isolate x:

-20x = 39

x = -39/20

Therefore, the solution to the equation is x = -39/20.

4. Verification

To verify the solution, substitute x = -39/20 back into the original equation. If both sides of the equation are equal, then the solution is correct.

In conclusion, by expanding, simplifying, and solving for x, we have determined that the solution to the equation (x-5)(x+5)-(x+3)^2+3(x-2)^2=(x+1)^2-(x+4)(x-4)+3x^2 is x = -39/20.