Question

Please help.
What are the solutions to the following system of equations?
y = x2 + 12x + 30
8x − y = 10

(−4, −2) and (2, 5)

(−2, −4) and (2, 5)

(−2, −4) and (5, 2)

No Real Solutions

Answer 1

@Jack1

Answer 2

be unique y in second equation, then plug in to first equation...
in seceon 8x - y = 10 -> y = 8x - 10
now plug in it to first equation and find x...
y = x2 + 12x + 30 -> 8x - 10 = x2 +12x + 30

got it lady? ;)

Answer 3

hey @raebaby420 , sorry bout the "issues" last time

anyway, as @mhmdrz91 said
use the second equation to find what y equals (in terms of x)
then sub in that result into the first equation (as per post above, cheers mhmd)

Answer 4

so you'll get:
8x - 10 = x^2 +12x + 30

now combine "like terms"
so 8x = x^2 + 12x + 30 +10
8x = x^2 + 12x + 40

continuing:
0 = x^2 + 12x -8x + 40
-->0 = x^2 + 4x + 40

Answer 5

0 = x^2 + 4x + 40

from here you've got two options:
1. the quadratic formula (google it, whay too long to type)

or
2. factor above equation
so: 0 = x^2 + 4x + 40
= (x + a ) ( x + b )
now solve for a and b

Answer 6

a * b = 40
ax + bx = 4x, so a + b = 4

now those 2 equations dont make any sense (as no 2 rational numbers both add to equal 4 and multiply to equal 40)
so answer is: no real solution (d)

Answer 7

imagine for a second that the first equation was: y = x^2 + 18x + 30
and the 2nd equation was: y - 4x = -10

so rearrange eqn 2:
y = 4x - 10

sub that into eqn 1

then you would get:
y = x^2 + 18x + 30
4x - 10 = x^2 + 18x + 30
4x = x^2 + 18x + 30 +10
4x = x^2 + 18x + 40
0 = x^2 + 14x + 40

--> 0 = x^2 + 14x + 40 IS an equation that is solvable via factoring

proof:
0 = x^2 + 14x + 40
0 = (x + a) ( x + b)
now (from FOIL) a * b must equal 40
and a + b must equal 14

so the numbers that match that are 4 and 10
a * b = 40 = 4 *10
a + b = 14 = 4 + 10

Answer 8

therefore:
0 = x^2 + 14x + 40
0 = (x + a) ( x + b)
0 = (x + 4) (x + 10)

so (x + 4) times (x + 10) = zero
now the only thing that multiplies to equal zero is...ZERO

so either (x + 4) = 0 [ as 0 times (x +10) = 0 ]
which would mean that x = -4
(x + 4) = 0
(-4 + 4) = 0
0 = 0 proof

OR
(x + 10) = 0 [ as 0 times (x + 4) = 0 ]
which would mean that x = -10
(x + 10) = 0
(-10 + 10) = 0
0 = 0 proof

so there are really 2 answers to "what is x?"

1. if x = -4, (using our new 2nd equation)
y - 4(x) = -10
y - 4 ( -4) = -10
y + (16) = -10
y = -10 -16
y = -26

giving coordinates of (-4, -26) [ as (x, y) ]

or if x = -10 (also using the 2nd equation)
y - 4(x) = -10
y - 4 (-10) = -10
y + 40 = -10
y = -50
giving coordinates of (-10, -50) [ as (x, y) ]

hope this helps?