sqrt(50)+sqrt(12)-4 sqrt(18)+4 sqrt(27)=?

To solve the equation \(\sqrt{50} + \sqrt{12} - c\sqrt{18} + c\sqrt{24} = 9\), we first simplify the square roots:<br /><br />\[<br />\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}<br />\]<br />\[<br />\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}<br />\]<br />\[<br />\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}<br />\]<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 />\[<br />5\sqrt{2} + 2\sqrt{3} - c \cdot 3\sqrt{2} + c \cdot 2\sqrt{6} = 9<br />\]<br /><br />Next, we group the terms with similar radicals:<br /><br />\[<br />(5\sqrt{2} - 3c\sqrt{2}) + 2\sqrt{3} + 2c\sqrt{6} = 9<br />\]<br /><br />Factor out \(\sqrt{2}\) from the first group:<br /><br />\[<br />\sqrt{2}(5 - 3c) + 2\sqrt{3} + 2c\sqrt{6} = 9<br />\]<br /><br />Now, isolate the constant term on the right side:<br /><br />\[<br />\sqrt{2}(5 - 3c) + 2\sqrt{3} + 2c\sqrt{6} = 9<br />\]<br /><br />To solve for \(c\), we need to isolate \(c\). Let's assume that the coefficients of \(\sqrt{2}\), \(\sqrt{3}\), and \(\sqrt{6}\) are such that the equation holds true. We can equate the coefficients of each term separately.<br /><br />First, let's isolate the \(\sqrt{2}\) term:<br /><br />\[<br />\sqrt{2}(5 - 3c) = 0<br />\]<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 />Next, let's check the coefficients of \(\sqrt{3}\) and \(\sqrt{6}\):<br /><br />\[<br />2\sqrt{3} = 0 \implies 2 = 0 \quad \text{(which is not possible)}<br />\]<br /><br />\[<br />2c\sqrt{6} = 0 \implies 2c = 0 \implies c = 0 \quad \text{(which is not possible)}<br />\]<br /><br />Since the coefficients of \(\sqrt{3}\) and \(\sqrt{6}\) cannot be zero, we conclude that there is no solution for \(c\) that satisfies the equation.<br /><br />Therefore, the equation \(\sqrt{50} + \sqrt{12} - c\sqrt{18} + c\sqrt{24} = 9\) has no solution for \(c\).