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

Question

UpStudy AI · Accepted Answer

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.
The solutions to the equation are $$ x = 4 $$ and $$ x = 7 $$.

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