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Solve the equation and check the solution. \[ \sqrt{29 x+53}-9=x \]

Ask by Schofield Li. in the United States
Feb 24,2025

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Answer

The solutions to the equation are \( x = 4 \) and \( x = 7 \).

Solution

We start with the equation   √(29x + 53) – 9 = x. Step 1. Isolate the square root by adding 9 to both sides:   √(29x + 53) = x + 9. Step 2. Determine the domain. The expression under the square root must be nonnegative, so:   29x + 53 ≥ 0 ⟹ x ≥ –53⁄29 ≈ –1.83. Also, since the left-hand side is nonnegative, the right-hand side must be as well:   x + 9 ≥ 0 ⟹ x ≥ –9. Thus, any solution must satisfy x ≥ –53⁄29. Step 3. Square both sides to eliminate the square root:   (√(29x + 53))² = (x + 9)²   29x + 53 = x² + 18x + 81. Step 4. Rearrange the equation to standard quadratic form:   0 = x² + 18x + 81 – 29x – 53    = x² – 11x + 28. Step 5. Factor the quadratic:   x² – 11x + 28 = (x – 4)(x – 7) = 0. Thus, the potential solutions are:   x = 4  or  x = 7. Step 6. Check each solution in the original equation to rule out any extraneous roots. For x = 4:   Left-hand Side = √(29(4) + 53) – 9 = √(116 + 53) – 9 = √169 – 9 = 13 – 9 = 4.   Right-hand Side = 4.   Since 4 = 4, x = 4 is valid. For x = 7:   Left-hand Side = √(29(7) + 53) – 9 = √(203 + 53) – 9 = √256 – 9 = 16 – 9 = 7.   Right-hand Side = 7.   Since 7 = 7, x = 7 is valid. Both solutions lie in the required domain. Final Answer:   The solutions to the equation are x = 4 and x = 7.

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To solve the equation \(\sqrt{29x + 53} - 9 = x\), we'll first isolate the square root term: \[ \sqrt{29x + 53} = x + 9 \] Next, we'll square both sides to eliminate the square root: \[ 29x + 53 = (x + 9)^2 \] Expanding the right-hand side: \[ 29x + 53 = x^2 + 18x + 81 \] Now, rearranging the equation: \[ 0 = x^2 + 18x + 81 - 29x - 53 \] This simplifies to: \[ 0 = x^2 - 11x + 28 \] Next, we can factor the quadratic equation: \[ 0 = (x - 7)(x - 4) \] Setting each factor equal to zero gives us the solutions: \[ x - 7 = 0 \quad \Rightarrow \quad x = 7 \] \[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \] Now we will check both solutions in the original equation. **Checking \(x = 7\):** \[ \sqrt{29(7) + 53} - 9 = 7 \] \[ \sqrt{203 + 53} - 9 = 7 \] \[ \sqrt{256} - 9 = 7 \] \[ 16 - 9 = 7 \] This is correct. **Checking \(x = 4\):** \[ \sqrt{29(4) + 53} - 9 = 4 \] \[ \sqrt{116 + 53} - 9 = 4 \] \[ \sqrt{169} - 9 = 4 \] \[ 13 - 9 = 4 \] This is also correct. Thus, the solutions to the equation \(\sqrt{29x + 53} - 9 = x\) are: \[ \boxed{7 \text{ and } 4} \]

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