Questions: Solve for (w), where (w) is a real number. [ sqrt5 w+14=w ] If there is more than one solution, separate them with commas. If there is no solution, click on "No solution". [ w= ]

Solution Steps

To solve the equation \(\sqrt{5w + 14} = w\), we first eliminate the square root by squaring both sides of the equation. This will result in a quadratic equation. We then rearrange the terms to form a standard quadratic equation and solve for \(w\) using the quadratic formula. Finally, we check the solutions in the original equation to ensure they are valid, as squaring can introduce extraneous solutions.

We start with the equation: \[ \sqrt{5w + 14} = w \]

To eliminate the square root, we square both sides: \[ 5w + 14 = w^2 \]

Rearranging the equation gives us: \[ w^2 - 5w - 14 = 0 \]

Using the quadratic formula \(w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = -5\), and \(c = -14\), we find: \[ w = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-14)}}{2 \cdot 1} = \frac{5 \pm \sqrt{25 + 56}}{2} = \frac{5 \pm \sqrt{81}}{2} \] This simplifies to: \[ w = \frac{5 \pm 9}{2} \] Thus, the solutions are: \[ w = \frac{14}{2} = 7 \quad \text{and} \quad w = \frac{-4}{2} = -2 \]

We need to check if these solutions satisfy the original equation:

1. For \(w = 7\): \[ \sqrt{5(7) + 14} = \sqrt{35 + 14} = \sqrt{49} = 7 \quad \text{(valid)} \]
2. For \(w = -2\): \[ \sqrt{5(-2) + 14} = \sqrt{-10 + 14} = \sqrt{4} = 2 \quad \text{(not valid, since } -2 \neq 2\text{)} \]

Final Answer