SOLUTION: For which values of P will x^2+p+x=3(x+1) have real roots

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Question 858686: For which values of P will x^2+p+x=3(x+1) have real roots
Found 3 solutions by mananth, LinnW, Theo:
Answer by mananth(16946)   (Show Source):
You can put this solution on YOUR website!
x^2+p+x=3(x+1)
x^2+p+x=3x+3
x^2-2x+(p-3)=0
For roots to be equal
the discriminant b^2-4ac =0 in the equation ax^2+bx+c=0
here a= 1, b= -2 c= (p-3)
-2^2 - 4*1*(p-3)=0
4-4p+12=0
4p=16
p=4

Answer by LinnW(1048)   (Show Source):
You can put this solution on YOUR website!

x^2 + p + x = 3x + 3
add -3x -3 to each side
x^2 -2x -3 + p
Determine whether roots are real by
seeing if where
a = 1 , b = -2 , c = ( -3 + p)

When - 4p + 16 > 0 we have real roots
add -16 to each side
-4p > -16
divide each side by -4
p < 4

Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website!
your solution is in the attached picture.

number 1 takes your original equation and converts it to standard form of a quadratic equation.
that form is ax^2 + bx + c = 0
once the equation is transformed, you get:
x^2 - 2x + (p-3) = 0 which results in:
a = 1
b = -2
c = (p - 3)

number 2 takes the discriminant from the quadratic formula that is used to perform the test to see if the roots are real or not.
if the discriminant is not negative, then the roots are real.
if the discriminant is negative, then the roots are not real.

the quadratic formula is:

-b +/- sqrt(b^2-4ac) x = ----------------------- 2a

the discriminant is the part that is under the square root sign.
the discriminant is equal to b^2 - 4ac.
if it is not negative the roots are real.
if it is negative, the roots are not real.
this is because the square root of a negative number is not real.

number 3 shows the discriminant equation that needs to be satisfied for the roots to be real.
that equation is b^2 - 4ac >= 0

number 4 takes the equation from number 3 and replaces b with (-2) and replaces a with 1 and replaces c with (p - 3) and solves the equation of b^2 - 4ac >= 0
the solution is that p must be smaller than or equal to 4 in order for the discriminant to be greater than or equal to 0.

number 5 takes the original equation of x^2 - 2x + (p - 3) = 0 and replaces p with 4 to see if the roots are real.
they are real.
this is good because 4 is smaller than or equal to 4 so the requirements are satisfied.

number 6 takes the original equation of x^2 - 2x + (p - 3) = 0 and replaces p with 3 to see if the roots are real.
they are real.
this is good because 3 is smaller than or equal to 4 so the requirements are again satisfied.

number 7 takes the original equation of x^2 - 2x + (p - 3) and replaces p with 5 to see if the roots are real.
they are not real.
this is good because 5 is not smaller than or equal to 4 so the requirements are again satisfied.

if p is smaller than or equal to 4, the roots are real.
if p is not smaller than or equal to 4, the roots are not real.