QUESTION 1
The first system is
[tex]y = {x}^{2} + 4x + 7[/tex]
and
[tex]y = 2[/tex]
Let us equate the two system to obtain,
[tex] {x}^{2} + 4x + 7 = 2[/tex]
Let us rewrite in general quadratic format to get
[tex] {x}^{2} + 4x + 7 - 2 = 0[/tex]
[tex] {x}^{2} + 4x + 5 = 0[/tex]
This implies that,
[tex]a=1,b=4, c=5[/tex]
We now find the discriminant,
[tex]D=b^2-4ac[/tex]
[tex]D=4^2-4(1)(5) = - 4[/tex]
Since the discriminant is negative, the system has no real number solution.
QUESTION 2
The second system is
[tex]y = {x}^{2} - 2[/tex]
and
[tex]y = x + 5[/tex]
We equate the two system to obtain,
[tex] {x}^{2} - 2 = x + 5[/tex]
This implies that,
[tex] {x}^{2} - x - 2 - 5= 0[/tex]
[tex] {x}^{2} - x - 7 = 0[/tex]
[tex]a=1,b=-1,c=-7[/tex]
[tex]D = {b}^{2} - 4ac[/tex]
We substitute the values to obtain,
[tex]D = {( - 1)}^{2} - 4(1)( - 7) = 29[/tex]
Since the discriminant is positive, the system has real number solutions.
QUESTION 3
The given system is
[tex]y = - {x}^{2} - 3[/tex]
[tex]y = 9 + 2x[/tex]
Equate the system to get,
[tex] - {x}^{2} - 3 = 9 + 2x[/tex]
Rewrite in general quadratic format
[tex] - {x}^{2} - 2x - 3 - 9= 0[/tex]
[tex] - {x}^{2} - 2x - 12= 0[/tex]
Divide through by -1 to get,
[tex] {x}^{2} + 2x + 12 = 0[/tex]
[tex]
a=1,b=2,c=12[/tex]
Using the discriminant we obtain,
[tex]D= {2}^{2} - 4(1)(12) = - 44[/tex]
Since the discriminant is negative the system has no real number solution.
QUESTION 4
The given system is,
[tex]y = - 3x - 6[/tex]
and
[tex]y = 2 {x}^{2} - 7x[/tex]
Equate the two equations to get,
[tex] 2 {x}^{2} - 7x = - 3x - 6[/tex]
Rewrite in general quadratic format, to obtain,
[tex] 2 {x}^{2} - 7x + 3x + 6 = 0[/tex]
[tex] 2 {x}^{2} - 4x + 6 = 0[/tex]
Divide through by 2 to obtain,
[tex] {x}^{2} - 2x + 3= 0[/tex]
This implies that,
[tex]a=1, b=-2,c=3[/tex]
We calculate the discriminant to obtain,
[tex]
D= {( - 2)}^{2} - 4(1)(3) = - 8[/tex]
Since the discriminant is negative the system has no real number solution.
QUESTION 5
The given system is
[tex]y = {x}^{2} [/tex]
and
[tex]y = 10 -8 x[/tex]
Equate the two equations to get,
[tex] {x}^{2} = 10 - 8x[/tex]
Rewrite in general quadratic format to obtain,
[tex] {x}^{2} + 8x - 10 = 0[/tex]
[tex]a=1,b=8,c=-10[/tex]
We now calculate the discriminant to get,
[tex] D= {8}^{2} - 4(1)( - 10) =104[/tex]
Since the discriminant is positive the system has real number solutions.
Answer:
A, C, D
Step-by-step explanation:
I took the test. The other persons was very long so i figured id help make it shorter and straight to the point. Hope this helps :)
