# Thread: parabolas and line intersection

1. ## parabolas and line intersection

Determine the equations of lines with slope 2 and that intersect the quadratic function f(x)=x(6-x)?

2. Originally Posted by mathhelp92
Determine the equations of lines with slope 2 and that intersect the quadratic function f(x)=x(6-x)?
let the lines be $y = 2x + k$, where $k$ is the y-intercept

$2x+k = x(6-x)
$

$2x+k = 6x - x^2$

$x^2 - 4x + k = 0$

to have real solutions, $b^2 - 4ac \geq 0$

$16 - 4k \geq 0$

$k \leq 4$

3. Originally Posted by skeeter
let the lines be $y = 2x + k$, where $k$ is the y-intercept

$2x+k = x(6-x)
$

$2x+k = 6x - x^2$

$x^2 - 4x + k = 0$

to have real solutions, $b^2 - 4ac \geq 0$

$16 - 4k \geq 0$

$k \leq 4$

THANK YOU...but this way I will get the roots of it
i need the equations for the line that have a slope of 2 and they pass through the parabola once, never n twice

4. Originally Posted by mathhelp92
THANK YOU...but this way I will get the roots of it
i need the equations for the line that have a slope of 2 and they pass through the parabola once, never n twice
To have 0 real solutions, the discriminant is negative. Solve for k.

To have 1 real solution, the discriminant is 0. Solve for k.

To have 2 real solutions, the discriminant is positive. Solve for k.

5. Originally Posted by Prove It
To have 0 real solutions, the discriminant is negative. Solve for k.

To have 1 real solution, the discriminant is 0. Solve for k.

To have 2 real solutions, the discriminant is positive. Solve for k.

So if k > 4, so then in the equation y=mx+b it will be y=2x+4..I THINK
so thats 2 roots

so how do we solve for only one root or no roots

because if we make both equations equal to each other, we will get 4 every time :s this is where i am confused

WHAT WE ARE TRYING TO SOLVE FOR IS THE VALUE OF B IN y=mx+b for 3 equations soo 3 different values of b

6. Originally Posted by mathhelp92
So if k > 4, so then in the equation y=mx+b it will be y=2x+4..I THINK
so thats 2 roots

so how do we solve for only one root or no roots

because if we make both equations equal to each other, we will get 4 every time :s this is where i am confused

WHAT WE ARE TRYING TO SOLVE FOR IS THE VALUE OF B IN y=mx+b for 3 equations soo 3 different values of b
If there aren't any intersection points then the value of $b$ is ANYTHING LESS THAN 4. So use a letter like $k$ and say that $k <4$.
If there is only one intersection point then the value of $b$ IS 4.
If there are two intersection points then the value of $b$ is ANYTHING GREATER THAN 4. So use a letter like $k$ and say that $k > 4$.