Questions: Find all real solutions of the equation. (Enter your answers as a comma-separated list.) x - sqrt(7-3 x) = 0 x = I / 2

Solution Steps

To solve the equation \( x - \sqrt{7 - 3x} = 0 \), we need to isolate \( x \). We can start by moving the square root term to the other side of the equation and then squaring both sides to eliminate the square root. After that, we solve the resulting quadratic equation for \( x \).

Given the equation: \[ x - \sqrt{7 - 3x} = 0 \]

We can isolate the square root term by moving it to the other side: \[ x = \sqrt{7 - 3x} \]

To eliminate the square root, we square both sides of the equation: \[ x^2 = 7 - 3x \]

Rearrange the equation to form a standard quadratic equation: \[ x^2 + 3x - 7 = 0 \]

Solve the quadratic equation using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1 \), \( b = 3 \), and \( c = -7 \).

The solutions are: \[ x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-7)}}{2 \cdot 1} \] \[ x = \frac{-3 \pm \sqrt{9 + 28}}{2} \] \[ x = \frac{-3 \pm \sqrt{37}}{2} \]

The solutions are: \[ x = \frac{-3 + \sqrt{37}}{2} \] \[ x = \frac{-3 - \sqrt{37}}{2} \]

Since we are looking for real solutions, we evaluate these expressions: \[ x = \frac{-3 + \sqrt{37}}{2} \approx 1.5414 \] \[ x = \frac{-3 - \sqrt{37}}{2} \approx -4.5414 \]

We need to verify which of these solutions satisfy the original equation \( x - \sqrt{7 - 3x} = 0 \).

For \( x \approx 1.5414 \): \[ 1.5414 - \sqrt{7 - 3 \cdot 1.5414} \approx 1.5414 - \sqrt{7 - 4.6242} \approx 1.5414 - \sqrt{2.3758} \approx 1.5414 - 1.5414 = 0 \]

For \( x \approx -4.5414 \): \[ -4.5414 - \sqrt{7 - 3 \cdot (-4.5414)} \approx -4.5414 - \sqrt{7 + 13.6242} \approx -4.5414 - \sqrt{20.6242} \] Since the square root of a positive number is positive, this does not satisfy the original equation.

Final Answer