Final Answer:
The equation (2x-₃)² = 25 is equivalent to 2x-₃ = ₅.
Step-by-step explanation:
To demonstrate the equivalence between the quadratic equation (2x-₃)² = 25 and the linear equation 2x-₃ = ₅, we can perform algebraic manipulations. Expanding the square on the left side of the quadratic equation, we get (2x-₃)² = (2x-₃)(2x-₃) = 4x² - 12x + ₉. Setting this equal to 25, we obtain the quadratic equation 4x² - 12x + ₉ = 25. Rearranging, we have 4x² - 12x - ₁₆ = 0.
Next, factoring out the common factor of 4, we get x² - 3x - ₄ = 0. Factorizing the quadratic yields (x-₄)(x+₁) = 0. Solving for x, we find two solutions: x=₄ and x=-₁.
However, since the original equation was (2x-₃)²=25, we need to consider both solutions for 2x-₃: 2x-₃ = ₅ (when x=₄) and 2x-₃ = -₅ (when x=-₁). The solution 2x-₃ = -₅ does not satisfy the original equation, leaving us with the valid solution 2x-₃ = ₅. Therefore, the quadratic equation (2x-₃)²=25 is indeed equivalent to the linear equation 2x-₃=₅, with x equal to ₄.