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Let’s check out your problem:

Let x=sqrt(7+sqrt(7+sqrt(7+)))

Let x=7+7+7+ x=\sqrt{7+\sqrt{7+\sqrt{7+}}}

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Full solution

Q. Let x=7+7+7+ x=\sqrt{7+\sqrt{7+\sqrt{7+}}}
  1. Recognize repeating pattern: First, let's recognize that the expression is a repeating pattern, so we can set xx to the entire expression. That means x=7+xx = \sqrt{7 + x}.
  2. Square both sides: Now, we square both sides to get rid of the square root. So, x2=7+xx^2 = 7 + x.
  3. Rearrange equation: Next, we rearrange the equation to set it to zero. We get x2−x−7=0x^2 - x - 7 = 0.
  4. Use quadratic formula: We can't factor this easily, so we'll use the quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where a=1a = 1, b=−1b = -1, and c=−7c = -7.
  5. Plug values into formula: Plugging the values into the quadratic formula, we get x=1±1+4×72x = \frac{1 \pm \sqrt{1 + 4 \times 7}}{2}.
  6. Simplify expression: Simplify the expression under the square root: x=1±1+282x = \frac{1 \pm \sqrt{1 + 28}}{2}.
  7. Add numbers under square root: Add the numbers under the square root: x=1±292x = \frac{1 \pm \sqrt{29}}{2}.
  8. Take positive square root: Since we're looking for a positive value of xx (because we can't have a negative length), we take the positive square root: x=(1+29)/2x = (1 + \sqrt{29}) / 2.

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