Question

if the system {y=X^2 +2x+3 y=kx-1 has exactly one solution, then what does k equal

Answer 1

Equate the coefficients since both are equal
x^2+2x+3=kx-1

power of x in 2x is 1 and power x in kx is 1
so k=2

Answer 2

@ipm1988 but how does that make it equal?

Answer 3

look at the equations both are equal to y

Answer 4

y=x^2+2x+3
and y =kx-1
so since both are equal to y therefore x^2+2x+3=kx-1

Answer 5

@ipm1988 then how would you get from that to k=2

Answer 6

so that has exactly one solution then take D=0 (D=discriminan)!

Answer 7

@kaileyhedgepeth are you familiar with the concept of comparing coefficients of variables of two equations

Answer 8

your second equation y=kx-2 can also be written as a quadratic equation even though it is a linear equation

Answer 9

@ipm1988 can you do this problem step by step please?

Answer 10

from x^2+2x+3=kx-1 or
x^2+2x+3-kx+1=0
x^2+(2-k)x+4=0
just make it D=0, solve for k! do u know what's that discriminan ?

Answer 11

@RadEn after that step what do you do and no i dont know what that is

Answer 12

well, if given a quadratic equation :
ax^2+bx+c=0,
so its discriminan value is
D=b^2-4ac (remember this)!
from info above, if given x^2+(2-k)x+4=0 (it means a=1, b=2-k, c=4)
so, D=(2-k)^2 - 4(1)(4)
D= 4-4k+k^2-16
D=k^2-4k-12
because D=0
therefor, k^2-4k-12=0 or
(k-6)(k+2)=0
zeros value of that factors is
k-6=0 --->k=6
k+2=0 ---->k=-2
so, k={-2,6}

Answer 13

@RadEn but there can only be one answer

Answer 14

yea, that's right :)
it means, if we subtitute k=-2 or k=6 then
y=X^2 +2x+3 and y=kx-1, both equation can has only one solution

Answer 15

so, the possible values of k are -2 or 6

Answer 16

do u understand ?

Answer 17

yes thanks!

Answer 18

yw