Solve each equation by completing the square. If necessary, round to the nearest hundredth. Please show all steps.
1. [tex]b^2 + 10b = 75[/tex]
2. [tex]n^2 - 20n = -75[/tex]
3. [tex]t^2 + 8t - 9 = 0[/tex]
4. [tex]m^2 - 2m - 8 = 0[/tex]
5. [tex]v^2 + 4v - 2 = 0[/tex]
Answer (2)
1. ) b 2 + 10 b = 75 b 2 + 10 b − 75 = 0 Δ = b 2 − 4 a c = 1 0 2 − 4 ∗ 1 ∗ ( 75 ) = 100 + 300 = 400 Δ = 400 = 20
b 1 = 2 a − b − Δ = 2 − 10 − 20 = 2 − 30 = − 15 b 2 = 2 a − b + Δ = 2 − 10 + 20 = 2 10 = 5
2. ) n 2 − 20 n = − 75 n 2 − 20 n + 75 = 0 Δ = b 2 − 4 a c = ( − 20 ) 2 − 4 ∗ 1 ∗ 75 = 400 − 300 = 100 Δ = 100 = 10
n 1 = 2 a − b − Δ = 2 20 − 10 = 2 10 = 5 2 = 2 a − b + Δ 2 20 + 10 = 2 30 = 15
3. ) t 2 + 8 t − 9 = 0 Δ = b 2 − 4 a c = 8 2 − 4 ∗ 1 ∗ ( − 9 ) = 64 + 36 = 100 Δ = 100 = 10
t 1 = 2 a − b − Δ = 2 − 8 − 10 = 2 − 18 = − 9 t 2 = 2 a − b + Δ 2 − 8 + 10 = 2 2 = 1
4. ) m 2 − 2 m − 8 = 0 Δ = b 2 − 4 a c = ( − 2 ) 2 − 4 ∗ 1 ∗ ( − 8 ) = 4 + 32 = 36 Δ = 36 = 6
m 1 = 2 a − b − Δ = 2 2 − 6 = 2 − 4 = − 2 m 2 = 2 a − b + Δ = 2 2 + 6 = 2 8 = 4
5. ) v 2 + 4 v − 2 = 0 Δ = b 2 − 4 a c = 4 2 − 4 ∗ 1 ∗ ( − 2 ) = 16 + 8 = 24 Δ = 24 = 4 ∗ 6 = 2 6
v 1 = 2 a − b − Δ = 2 − 4 − 2 6 = 2 2 ( − 2 − 6 = − 2 − 6 ≈ − 2 − 2 , 45 ≈ − 4 , 45 v 2 = 2 a − b + Δ = 2 − 4 + 2 6 = 2 2 ( 6 − 2 = 6 − 2 ≈ 2 , 45 − 2 ≈ 0 , 45
By completing the square for each equation, we find solutions for each variable. The solutions for each equation are as follows: b : − 15 , 5 , n : 5 , 15 , t : − 9 , 1 , m : − 2 , 4 , and v : − 4.45 , 0.45 .
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