What is the largest number c such that 2x^2 + 5x + c = -x^2 + 12x + 18 has at least one real solution? Express your answer as a common fraction.
+14905 What is the largest number c?
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\(2x^2 + 5x + c = -x^2 + 12x + 18\\ 3x^2-7x-18+c=0\ |\ /3\\ x^2- \frac{7}{3}x+\frac{c-18}{3}=0\\ x=\frac{7}{6}\pm \sqrt{ \frac{ 49}{36}-\frac{c-18}{3} }\)
\(\frac{ 49}{36}-\frac{c-18}{3}\geq 0\\ 147-36c+648\geq 0\\ 36c\leq 795\\ \color{blue}c\leq 22\frac{1}{12}\)
\(The\ largest\ number\ c\ is\ \color{blue}22\frac{1}{12}.\)
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