(x+1)^2-(x-2)^2=(x+3)^2+x^2-20

Jasonbradley

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Jun 16, 2024

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Solving the Equation: (x+1)^2-(x-2)^2=(x+3)^2+x^2-20

This article explores the process of solving the given equation: (x+1)^2-(x-2)^2=(x+3)^2+x^2-20. We will utilize algebraic manipulation to isolate the variable 'x' and find its solution.

Expanding and Simplifying

Firstly, we expand the squares on both sides of the equation using the formula (a+b)^2 = a^2 + 2ab + b^2.

(x+1)^2-(x-2)^2=(x+3)^2+x^2-20

x^2 + 2x + 1 - (x^2 - 4x + 4) = x^2 + 6x + 9 + x^2 - 20

Next, we distribute the negative sign on the left side and combine like terms.

x^2 + 2x + 1 - x^2 + 4x - 4 = 2x^2 + 6x - 11

6x - 3 = 2x^2 + 6x - 11

Isolating the Variable

Now, we rearrange the equation to set it equal to zero.

0 = 2x^2 - 11 + 3

0 = 2x^2 - 8

Next, we divide both sides by 2.

0 = x^2 - 4

Finally, we factor the equation.

0 = (x + 2)(x - 2)

Finding the Solution

Setting each factor equal to zero, we solve for 'x'.

x + 2 = 0 or x - 2 = 0

x = -2 or x = 2

Therefore, the solutions to the equation (x+1)^2-(x-2)^2=(x+3)^2+x^2-20 are x = -2 and x = 2.