sqrt(50)+sqrt(12)-4 sqrt(18)+4 sqrt(23)=?

To solve the equation \(\sqrt{50} + \sqrt{12} - c\sqrt{18} + c\sqrt{24} = 0\), we first simplify the square roots:<br /><br />\[<br />\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}<br />\]<br /><br />\[<br />\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}<br />\]<br /><br />\[<br />\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}<br />\]<br /><br />\[<br />\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}<br />\]<br /><br />Substituting these simplified forms into the equation, we get:<br /><br />\[<br />5\sqrt{2} + 2\sqrt{3} - c \cdot 3\sqrt{2} + c \cdot 2\sqrt{6} = 0<br />\]<br /><br />Next, we group the terms with similar radicals:<br /><br />\[<br />(5\sqrt{2} - 3c\sqrt{2}) + 2\sqrt{3} + 2c\sqrt{6} = 0<br />\]<br /><br />Factor out the common radicals:<br /><br />\[<br />\sqrt{2}(5 - 3c) + 2\sqrt{3} + 2c\sqrt{6} = 0<br />\]<br /><br />For the equation to be zero, each term must be zero. Therefore, we set each part equal to zero:<br /><br />1. \(\sqrt{2}(5 - 3c) = 0\)<br /><br />Since \(\sqrt{2} \neq 0\), we have:<br /><br />\[<br />5 - 3c = 0 \implies 3c = 5 \implies c = \frac{5}{3}<br />\]<br /><br />2. \(2\sqrt{3} = 0\)<br /><br />This term is already zero, so it does not provide any new information.<br /><br />3. \(2c\sqrt{6} = 0\)<br /><br />Since \(\sqrt{6} \neq 0\), we have:<br /><br />\[<br />2c = 0 \implies c = 0<br />\]<br /><br />However, \(c = 0\) does not satisfy the first equation. Therefore, the only solution is:<br /><br />\[<br />c = \frac{5}{3}<br />\]<br /><br />Thus, the value of \(c\) that satisfies the equation is:<br /><br />\[<br />\boxed{\frac{5}{3}}<br />\]