\( 1 \leftarrow \quad \) Solve. \( \quad x-\sqrt{10-9 x}=-20 \)

To solve the equation \( x - \sqrt{10 - 9x} = -20 \), first, move \( x \) to the right side: \[ \sqrt{10 - 9x} = x + 20 \] Next, square both sides to eliminate the square root: \[ 10 - 9x = (x + 20)^2 \] Expanding the right side: \[ 10 - 9x = x^2 + 40x + 400 \] Rearranging all terms to one side gives: \[ 0 = x^2 + 49x + 390 \] Now, use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): Here, \( a = 1 \), \( b = 49 \), and \( c = 390 \). Calculating the discriminant: \[ b^2 - 4ac = 49^2 - 4 \cdot 1 \cdot 390 = 2401 - 1560 = 841 \] Now plug this into the quadratic formula: \[ x = \frac{-49 \pm \sqrt{841}}{2 \cdot 1} = \frac{-49 \pm 29}{2} \] Calculating the two possible values for \( x \): 1. \( x = \frac{-49 + 29}{2} = \frac{-20}{2} = -10 \) 2. \( x = \frac{-49 - 29}{2} = \frac{-78}{2} = -39 \) Now, we need to verify which solutions are valid by substituting them back into the original equation. For \( x = -10 \): \[ -10 - \sqrt{10 - 9(-10)} = -10 - \sqrt{10 + 90} = -10 - \sqrt{100} = -10 - 10 = -20 \quad \text{(Valid)} \] For \( x = -39 \): \[ -39 - \sqrt{10 - 9(-39)} = -39 - \sqrt{10 + 351} = -39 - \sqrt{361} = -39 - 19 = -58 \quad \text{(Invalid)} \] Thus, the valid solution is: \[ \boxed{-10} \]