SOLUTION: 1. Determine whether the following equations have a solution or not? Justify your answer. a) x^2 + 6x - 7 = 0 b) z^2 + z + 1 = 0 c) (3)^(1/2)*y^2 - 4y - 7*(3)^(1/2) = 0

Algebra ->  Graphs -> SOLUTION: 1. Determine whether the following equations have a solution or not? Justify your answer. a) x^2 + 6x - 7 = 0 b) z^2 + z + 1 = 0 c) (3)^(1/2)*y^2 - 4y - 7*(3)^(1/2) = 0       Log On


   

Question 155165: 1. Determine whether the following equations have a solution or not? Justify your answer.

a) x^2 + 6x - 7 = 0
b) z^2 + z + 1 = 0
c) (3)^(1/2)*y^2 - 4y - 7*(3)^(1/2) = 0
d) 2x^2 - 10x + 25 = 0
e) 2x^2 - 6x + 5 = 0
f) s^2 - 4s + 4 = 0
g) (5/6)x^2 - 7x - 6/5 = 0
h) 7a^2 + 8a + 2 = 0

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
a)


From x%5E2%2B6x-7 we can see that a=1, b=6, and c=-7


D=b%5E2-4ac Start with the discriminant formula.


D=%286%29%5E2-4%281%29%28-7%29 Plug in a=1, b=6, and c=-7


D=36-4%281%29%28-7%29 Square 6 to get 36


D=36--28 Multiply 4%281%29%28-7%29 to get %284%29%28-7%29=-28


D=36%2B28 Rewrite D=36--28 as D=36%2B28


D=64 Add 36 to 28 to get 64


Since the discriminant is greater than zero, this means that there are two real solutions.

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b)


From z%5E2%2Bz%2B1 we can see that a=1, b=1, and c=1


D=b%5E2-4ac Start with the discriminant formula.


D=%281%29%5E2-4%281%29%281%29 Plug in a=1, b=1, and c=1


D=1-4%281%29%281%29 Square 1 to get 1


D=1-4 Multiply 4%281%29%281%29 to get %284%29%281%29=4


D=-3 Subtract 4 from 1 to get -3


Since the discriminant is less than zero, this means that there are two complex solutions. In other words, there are no real solutions.

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c)

Start with the given expression


sqrt%283%29y%5E2-4y-7%2Asqrt%283%29 Rewrite as sqrt%283%29


From sqrt%283%29y%5E2-4y-7%2Asqrt%283%29, we can see that a=sqrt%283%29, b=-4, and c=7%2Asqrt%283%29


D=b%5E2-4ac Start with the discriminant formula

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D=100 Add

Since the discriminant is greater than zero, this means that there are two real solutions.


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d)



From 2x%5E2-10x%2B25 we can see that a=2, b=-10, and c=25


D=b%5E2-4ac Start with the discriminant formula.


D=%28-10%29%5E2-4%282%29%2825%29 Plug in a=2, b=-10, and c=25


D=100-4%282%29%2825%29 Square -10 to get 100


D=100-200 Multiply 4%282%29%2825%29 to get %288%29%2825%29=200


D=-100 Subtract 200 from 100 to get -100


Since the discriminant is less than zero, this means that there are two complex solutions. In other words, there are no real solutions.

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e)


From 2x%5E2-6x%2B5 we can see that a=2, b=-6, and c=5


D=b%5E2-4ac Start with the discriminant formula.


D=%28-6%29%5E2-4%282%29%285%29 Plug in a=2, b=-6, and c=5


D=36-4%282%29%285%29 Square -6 to get 36


D=36-40 Multiply 4%282%29%285%29 to get %288%29%285%29=40


D=-4 Subtract 40 from 36 to get -4


Since the discriminant is less than zero, this means that there are two complex solutions. In other words, there are no real solutions.

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f)



From s%5E2-4s%2B4 we can see that a=1, b=-4, and c=4


D=b%5E2-4ac Start with the discriminant formula.


D=%28-4%29%5E2-4%281%29%284%29 Plug in a=1, b=-4, and c=4


D=16-4%281%29%284%29 Square -4 to get 16


D=16-16 Multiply 4%281%29%284%29 to get %284%29%284%29=16


D=0 Subtract 16 from 16 to get 0


Since the discriminant is equal to zero, this means that there is one real solution.

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g)


From %285%2F6%29x%5E2-7x-%286%2F5%29 we can see that a=5%2F6, b=-7, and c=-6%2F5


D=b%5E2-4ac Start with the discriminant formula.


D=%28-7%29%5E2-4%285%2F6%29%28-6%2F5%29 Plug in a=5%2F6, b=-7, and c=-6%2F5


D=49-4%285%2F6%29%28-6%2F5%29 Square -7 to get 49


D=49--4 Multiply 4%285%2F6%29%28-6%2F5%29 to get %2810%2F3%29%28-6%2F5%29=-4


D=49%2B4 Rewrite D=49--4 as D=49%2B4


D=53 Add 49 to 4 to get 53


Since the discriminant is greater than zero, this means that there are two real solutions.

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h)


From 7a%5E2%2B8a%2B2 we can see that a=7, b=8, and c=2


D=b%5E2-4ac Start with the discriminant formula.


D=%288%29%5E2-4%287%29%282%29 Plug in a=7, b=8, and c=2


D=64-4%287%29%282%29 Square 8 to get 64


D=64-56 Multiply 4%287%29%282%29 to get %2828%29%282%29=56


D=8 Subtract 56 from 64 to get 8


Since the discriminant is greater than zero, this means that there are two real solutions.