# question

• Sep 24th 2009, 02:12 AM
mark
question
i'm not actually sure what this question is asking me but i've answered the main part of it. it goes:

a straight line has equation $y = 4x + k$ where k is a constant, and a parabola has equation $y = 3x^2 + 12x + 7$. show that the x coordinate of any points of intersection of the line and the parabola satisfy $3x^2 + 8x + 7 - k = 0$ hence find the range of values of k for which the line and parabola do not intersect.

i got the second bit and came up with $k < \frac{5}{3}$ which i'm sure is right, but i don't understand what its asking me before it says "hence find the range of values etc"

thanks
• Sep 24th 2009, 02:34 AM
Prove It
Quote:

Originally Posted by mark
i'm not actually sure what this question is asking me but i've answered the main part of it. it goes:

a straight line has equation $y = 4x + k$ where k is a constant, and a parabola has equation $y = 3x^2 + 12x + 7$. show that the x coordinate of any points of intersection of the line and the parabola satisfy $3x^2 + 8x + 7 - k = 0$ hence find the range of values of k for which the line and parabola do not intersect.

i got the second bit and came up with $k < \frac{5}{3}$ which i'm sure is right, but i don't understand what its asking me before it says "hence find the range of values etc"

thanks

The line and quadratic will intersect where the functions are equal.

So $4x + k = 3x^2 + 12x + 7$

$0 = 3x^2 + 8x + 7 - k$.

To show the values of $k$ for which there are not any intersections, we need to remember that the discriminant will be negative.

$\Delta = 8^2 - 4(3)(7 - k) < 0$

$64 - 84 + 12k < 0$

$-20 + 12k < 0$

$12k < 20$

$k < \frac{5}{3}$.
• Sep 24th 2009, 02:50 AM
mark
the bit i was looking for an explanation for was- "a straight line has equation $y = 4x + k$ where k is a constant, and a parabola has equation $y = 3x^2 + 12x + 7$ show that the x coordinate of any points of intersection of the line and the parabola satisfy $3x^2 + 8x + 7 - k = 0$"

• Sep 24th 2009, 02:51 AM
Prove It
Yes I did.

The two functions will be equal at the points of intersection, so

$4x + k = 3x^2 + 12x + 7$

and collecting everything on the one side gives

$0 = 3x^2 + 8x + 7 - k$

as required.