MODIFIED: If variable_94² + variable_94² + variable_94² = variable_94*variable_94 + variable_94*variable_94 + variable_94*variable_94, prove that variable_94 = variable_94 =

Step 1: Rearrange the equation

The first step is to move all terms to one side of the equation, setting it equal to zero. This standard algebraic manipulation helps us prepare the equation for further simplification and analysis.

Step 2: Multiply by 2

To facilitate the formation of perfect square trinomials, we multiply the entire equation by 2. This operation does not change the validity of the equation since multiplying both sides by a non-zero constant maintains equality.

Step 3: Group terms to form squares

Now, we regroup the terms to form three perfect square trinomials. We use the identity $(x-y)^2 = x^2 - 2xy + y^2$. Each group corresponds to the square of the difference between two variables.

Step 4: Factor the perfect squares

By factoring each perfect square trinomial, we simplify the equation into a sum of three squared terms. The square of any real number is always non-negative.

Step 5: Deduce the equality

Since the sum of three non-negative terms is zero, each term must individually be zero. This implies that $a-b=0$, $b-c=0$, and $c-a=0$. From these equalities, we can conclude that $a=b=c$.

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