Solving the Equation ( sqrt{x^2-7}2x-7 ) and Ensuring Real and Meaningful Solutions

Let's consider the equation ( sqrt{x^2-7}2x-7 ). We will solve this step by step and ensure that the solutions are real and meaningful.

Step-by-Step Solution

Step 1: Square Both Sides of the Equation

To eliminate the square root, we square both sides of the equation:

[ sqrt{x^2-7}2x-7 ] Squaring both sides, we get:

[ x^2-7(2x-7)^2 ]

This expands to:

[ x^2-74x^2-28x 49 ]

Rearranging all terms to one side, we get:

[ x^2-7-4x^2 28x-490 ]

Transforming the Equation

Combining like terms, we obtain:

[ -3x^2 28x-560 ]

To make the leading coefficient 1, we divide the entire equation by -3:

[ x^2-frac{28}{3}x frac{56}{3}0 ]

Next, we complete the square. To do this:

Divide the coefficient of ( x ) by 2 and square it: ( left(frac{-28}{3} times frac{1}{2}right)^2 left(frac{-14}{3}right)^2 frac{196}{9} ) Add and subtract this square inside the equation: ( x^2 - frac{28}{3}x frac{196}{9} - frac{196}{9} frac{56}{3} 0 ) Group the perfect square trinomial and combine constants: ( (x - frac{14}{3})^2 - frac{196}{9} frac{168}{9} 0 ) ( (x - frac{14}{3})^2 - frac{28}{9} 0 ) ( (x - frac{14}{3} frac{2sqrt{7}}{3})(x - frac{14}{3} - frac{2sqrt{7}}{3}) 0 ) ( x - frac{14}{3} - frac{2sqrt{7}}{3} 0 ) or ( x - frac{14}{3} frac{2sqrt{7}}{3} 0 ) ( x frac{14 2sqrt{7}}{3} ) or ( x frac{14 - 2sqrt{7}}{3} )

So, the potential solutions are:

[ x frac{14 pm 2sqrt{7}}{3} ]

Verifying the Solutions

Next, we need to check which of these solutions are valid by substituting them back into the original equation:

Solution 1: ( x frac{14 2sqrt{7}}{3} )

First, we need to check if this value satisfies the condition that the square root is defined and non-negative:

[ 2x - 7 2 left(frac{14 2sqrt{7}}{3}right) - 7 frac{28 4sqrt{7} - 21}{3} frac{7 4sqrt{7}}{3} ge 0 ]

Since ( x frac{14 2sqrt{7}}{3} ge frac{7}{2} ), both conditions are satisfied.

Solution 2: ( x frac{14 - 2sqrt{7}}{3} )

Similarly, we need to verify:

[ 2x - 7 2 left(frac{14 - 2sqrt{7}}{3}right) - 7 frac{28 - 4sqrt{7} - 21}{3} frac{7 - 4sqrt{7}}{3} le 0 ]

This value does not satisfy the non-negative condition for the square root.

Therefore, the only valid solution is:

[ x frac{14 2sqrt{7}}{3} ]

Conclusion

The solution to the equation ( sqrt{x^2-7}2x-7 ) is:

[ boxed{x frac{14 2sqrt{7}}{3}} ]