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Prove that the equation (2x-3)^2=25 is equivalent to 2x-3=5.

User Sgonzalez
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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.

User Brnunes
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