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.
2x2 + 5x + c = –x2 + 12x + 18 —> 3x2 – 7x + (c – 18) = 0
To have at least one real solution,
the discriminate must be > 0.
b2 – 4ac > 0
(–7)2 – (4)(3)(c – 18) > 0
49 – 12c + 216 > 0
–12c + 265 > 0
–12c > –265
12c < 265 (note that the inequality sign flips)
c < 265 / 12
c < 221/12
check answer
if c = 22, then b2 – 4ac = (–7)2 – (4)(3)(22 – 18)
49 – (4)(3)(4)
49 – 48 = 1 —> discriminant positive good
if c = 23, then b2 – 4ac = (–7)2 – (4)(3)(23 – 15)
49 – (4)(3)(5)
49 – 60 = –11 —> discriminant negative bad
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