Tính √(7 - x) + √(x - 5) = x² - 12x + 38

To solve the equation √(7-x) + √(x-5) = x²-12x+38, we can start by isolating one of the square roots on one side of the equation.

Let's isolate √(7-x):

√(7-x) = x²-12x+38 - √(x-5)

Now, square both sides of the equation to eliminate the square root:

(√(7-x))² = (x²-12x+38 - √(x-5))²

Simplifying the left side:

7-x = x⁴ - 24x³ + 2x²(19 - 2√(x-5)) + 2x(12√(x-5) - 38) + (x-5)

Expanding the right side:

7-x = x⁴ - 24x³ + 38x² - 4x²√(x-5) + 76x√(x-5) - 76√(x-5) + x - 5

Combining like terms:

0 = x⁴ - 24x³ + 38x² - 4x²√(x-5) + 76x√(x-5) - 76√(x-5) - 6x

Now, let's isolate the square root term:

4x²√(x-5) - 76x√(x-5) + 76√(x-5) = x⁴ - 24x³ + 38x² - 6x - x + 5

Factoring out the common factor of √(x-5):

√(x-5)(4x² - 76x + 76) = x⁴ - 24x³ + 38x² - 7x + 5

Now, let's simplify the expression inside the parentheses:

4x² - 76x + 76 = 4(x² - 19x + 19)

Substituting this back into the equation:

√(x-5)(4(x² - 19x + 19)) = x⁴ - 24x³ + 38x² - 7x + 5

Now, let's square both sides of the equation to eliminate the square root:

(x-5)(4(x² - 19x + 19))² = (x⁴ - 24x³ + 38x² - 7x + 5)²

Expanding both sides:

(x-5)(4(x² - 19x + 19))² = x⁸ - 48x⁷ + 76x⁶ - 14x⁵ + 10x⁴ - 48x³ + 76x² - 14x + 10

Multiplying out the squared term:

(x-5)(4(x² - 19x + 19))(4(x² - 19x + 19)) = x⁸ - 48x⁷ + 76x⁶ - 14x⁵ + 10x⁴ - 48x³ + 76x² - 14x + 10

Simplifying both sides:

4(x-5)(x² - 19x + 19)² = x⁸ - 48x⁷ + 76x⁶ - 14x⁵ + 10x⁴ - 48x³ + 76x² - 14x + 10

Now, we have a polynomial equation of degree 8. Unfortunately, there is no simple algebraic method to solve this equation. It may be necessary to use numerical methods or a graphing calculator to approximate the solutions.