3 x 2 + 12 x − 1 a = 3 ; b = 12 ; c = − 1 Δ = b 2 − 4 a c → Δ = 1 2 2 − 4 ⋅ 3 ⋅ ( − 1 ) = 144 + 12 = 156 x = 2 a − b ± Δ Δ = 156 = 4 ⋅ 39 = 4 ⋅ 39 = 2 39 x = 2 ⋅ 3 − 12 ± 2 39 = 3 − 6 ± 39 3 x 2 + 12 x − 1 = 3 ( x − 3 − 6 − 39 ) ( x − 3 − 6 + 39 ) = 3 ( x + 3 6 + 39 ) ( x + 3 6 − 39 )
To complete the square for 3 x 2 + 12 x − 1 , first factor out 3 from the x 2 and x terms, then add and subtract half the coefficient of x squared inside the parentheses. The final expression in completed square form is 3 ( x + 2 ) 2 − 13 .
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