Question

Solve each equation by completing the square. If necessary, round to the nearest hundredth.

Please show steps, steps needs.
1. b^2+10b=75
2. n^2-20n=-75
3.. t^2+8t-9=0
4. m^2-2m-8=0
5. v^2+4v-2=0

Answer 1

`1.)\\ \\ b^2+10b=75\\ \\ b^2+10b-75=0\\ \\\Delta = b^(2)-4ac = 10^(2)-4*1*(75)=100+300=400 \\ \\√(\Delta )=√(400)=20`


`b_(1)=(-b-√(\Delta ))/(2a) =(-10-20)/(2)=(-30)/(2)=-15\\ \\b_(2)=(-b+√(\Delta ))/(2a) =(-10+20)/(2)=(10)/(2)=5`




`2.)\\ \\ n^2-20n=-75\\ \\ n^2-20n+75=0\\ \\\Delta = b^(2)-4ac = (-20)^(2)-4*1*75=400-300=100 \\ \\√(\Delta )=√(100)=10`


`n_(1)=(-b-√(\Delta ))/(2a) =(20-10)/(2)=(10)/(2)=5\\ \\n_(2)=(-b+√(\Delta ))/(2a) (20+10)/(2)=(30)/(2)=15`




`3.)\\ \\t^2+8t-9=0\\ \\\Delta = b^(2)-4ac = 8^(2)-4*1* (-9)=64+36=100 \\ \\√(\Delta )=√(100)=10`


`t_(1)=(-b-√(\Delta ))/(2a) =(-8-10)/(2)=(-18)/(2)=-9\\ \\t_(2)=(-b+√(\Delta ))/(2a) (-8+10)/(2)=(2)/(2)=1`



`4.)\\ \\m^2-2m-8=0\\ \\\Delta = b^(2)-4ac = (-2)^(2)-4*1* (-8)=4+32=36\\ \\√(\Delta )=√(36)=6`


`m_(1)=(-b-√(\Delta ))/(2a) =(2-6)/(2)=(-4)/(2)=-2\\ \\m_(2)=(-b+√(\Delta ))/(2a)= (2+6)/(2)=(8)/(2)=4`




`5.)\\ \\ v^2+4v-2=0\\ \\\Delta = b^(2)-4ac = 4^(2)-4*1* (-2)=16+8=24\\ \\√(\Delta )=√(24)= √(4*6)=2√(6)`


`v_(1)=(-b-√(\Delta ))/(2a) =(-4-2√(6))/(2)=(2(-2-√(6))/(2)= -2-√(6)\approx -2-2,45\approx -4,45\\ \\v_(2)=(-b+√(\Delta ))/(2a)= (-4+2√(6))/(2)=(2(√(6)-2)/(2)= √(6)-2 \approx 2,45-2\approx 0,45`