Quadratic

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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