Questions: 3 √x = x - 5

Solution Steps

To solve the equation \(3 \sqrt{x} = x - 5\), we can isolate the square root term and then square both sides to eliminate the square root. This will result in a quadratic equation, which we can solve using the quadratic formula or by factoring.

Starting with the equation \(3 \sqrt{x} = x - 5\), we can rearrange it to isolate the square root term: \[ 3 \sqrt{x} = x - 5 \]

Next, we square both sides to eliminate the square root: \[ (3 \sqrt{x})^2 = (x - 5)^2 \] This simplifies to: \[ 9x = x^2 - 10x + 25 \]

Rearranging the equation gives us: \[ x^2 - 19x + 25 = 0 \]

Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = -19\), and \(c = 25\): \[ x = \frac{19 \pm \sqrt{(-19)^2 - 4 \cdot 1 \cdot 25}}{2 \cdot 1} \] Calculating the discriminant: \[ x = \frac{19 \pm \sqrt{361 - 100}}{2} = \frac{19 \pm \sqrt{261}}{2} \] This results in: \[ x = \frac{19 \pm 3\sqrt{29}}{2} \]

The solutions are: \[ x_1 = \frac{19 + 3\sqrt{29}}{2}, \quad x_2 = \frac{19 - 3\sqrt{29}}{2} \] Since \(x\) must be non-negative (as it is under a square root), we only consider \(x_1\).

Final Answer