2 m 2 − 7 m − 3 = 0 a = 2 ; b = − 7 ; c = − 3 Δ = b 2 − 4 a c → Δ = ( − 7 ) 2 − 4 ⋅ 2 ⋅ ( − 3 ) = 49 + 24 = 73 x = 2 a − b ± Δ → x = 2 ⋅ 2 7 ± 73 = 4 7 ± 73
4 b 2 + 8 b + 7 = 4 4 b 2 + 8 b + 7 − 4 = 0 4 b 2 + 8 b + 3 = 0 4 b 2 + 2 b + 6 b + 3 = 0 2 b ( 2 b + 1 ) + 3 ( 2 b + 1 ) = 0 ( 2 b + 1 ) ( 2 b + 3 ) = 0 ⟺ 2 b + 1 = 0 or 2 b + 3 = 0 2 b = − 1 or 2 b = − 3 b = − 2 1 or b = − 2 3
5 80 a 2 = 5 80 ⋅ a 2 = 5 16 ⋅ 5 ⋅ ∣ a ∣ = 5 ⋅ 16 ⋅ 5 ⋅ ∣ a ∣ = 5 ⋅ 4 ⋅ 5 ⋅ ∣ a ∣ = 20∣ a ∣ 5
− 6 150 r = − 6 25 ⋅ 6 r = − 6 25 ⋅ 6 r = − 6 ⋅ 5 6 r = − 30 6 r
3 2 = 3 2 ⋅ 3 3 = 3 2 3
5 3 6 = 5 1 ⋅ 3 6 = 5 1 3 6 = 5 1 2 = 5 2
To solve the first two equations, use the quadratic formula and factorization. The simplifications involve extracting square roots and rationalizing denominators. Each step follows algebraic rules to arrive at simplified forms or solutions.
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