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

To solve the equation \( x - \sqrt{10 - 9x} = -20 \), start by isolating the square root: \[ \sqrt{10 - 9x} = x + 20 \] Next, square both sides to eliminate the square root: \[ 10 - 9x = (x + 20)^2 \] Expanding the right side gives: \[ 10 - 9x = x^2 + 40x + 400 \] Now, rearranging the equation leads to: \[ 0 = x^2 + 49x + 390 \] This is a quadratic equation in standard form. You can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a = 1, b = 49, c = 390 \): Calculating the discriminant: \[ b^2 - 4ac = 49^2 - 4(1)(390) = 2401 - 1560 = 841 \] Now, since 841 is a perfect square, we can find \( x \): \[ x = \frac{-49 \pm \sqrt{841}}{2} = \frac{-49 \pm 29}{2} \] This gives two potential solutions: \[ x_1 = \frac{-20}{2} = -10, \quad x_2 = \frac{-78}{2} = -39 \] Now, check both solutions in the original equation to verify: For \( x = -10 \): \[ -10 - \sqrt{10 - 9(-10)} = -10 - \sqrt{100} = -10 - 10 = -20 \quad (\text{Valid}) \] For \( x = -39 \): \[ -39 - \sqrt{10 - 9(-39)} = -39 - \sqrt{360 + 10} = -39 - \sqrt{370} \quad (\text{Not valid since it doesn't equal -20}) \] So, the only solution is \( x = -10 \).