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07 Aug 2025, 08:56
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If \(\sqrt{2x-3}=x-3,\) then what is the value of x.
16 Aug 2025, 04:05
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Official Explanation
Square both sides of the given equation, in order to get rid of the square root:
Be careful on the right-side to square the binomial by using the F.O.I.L. method or the formula \((a - b)^2 = a^2 - 2ab + b^2.\) This yields \(2x - 3 = x^2 - 6x + 9\). To solve a quadratic, or x^2, equation, set one side equal to zero first, by grouping all the terms onto the other side. Here, subtract 2x and add 3 to both sides to obtain: 0 = \(x^2 - 8x + 12.\) Finally, factor the right side by noting that \(-6× -2\) equals 12, while \(-6 + -2 = -8.\) So \(0 = (x-2)(x-6),\) and x = 2 or x = 6. However, squaring both sides may produce an “extraneous” solution, so make sure to check each result in the original equation. If x = 2, the right side equals but the left equals 2 - 3 = -1. Only x = 6 satisfies the original equation, making both sides equal to 3.
Answer: 6
Square both sides of the given equation, in order to get rid of the square root:
Be careful on the right-side to square the binomial by using the F.O.I.L. method or the formula \((a - b)^2 = a^2 - 2ab + b^2.\) This yields \(2x - 3 = x^2 - 6x + 9\). To solve a quadratic, or x^2, equation, set one side equal to zero first, by grouping all the terms onto the other side. Here, subtract 2x and add 3 to both sides to obtain: 0 = \(x^2 - 8x + 12.\) Finally, factor the right side by noting that \(-6× -2\) equals 12, while \(-6 + -2 = -8.\) So \(0 = (x-2)(x-6),\) and x = 2 or x = 6. However, squaring both sides may produce an “extraneous” solution, so make sure to check each result in the original equation. If x = 2, the right side equals but the left equals 2 - 3 = -1. Only x = 6 satisfies the original equation, making both sides equal to 3.
Answer: 6
