1. ## 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 $\displaystyle y = 4x + k$ where k is a constant, and a parabola has equation $\displaystyle y = 3x^2 + 12x + 7$. show that the x coordinate of any points of intersection of the line and the parabola satisfy $\displaystyle 3x^2 + 8x + 7 - k = 0$ 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 $\displaystyle k < \frac{5}{3}$ 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

2. 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 $\displaystyle y = 4x + k$ where k is a constant, and a parabola has equation $\displaystyle y = 3x^2 + 12x + 7$. show that the x coordinate of any points of intersection of the line and the parabola satisfy $\displaystyle 3x^2 + 8x + 7 - k = 0$ 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 $\displaystyle k < \frac{5}{3}$ 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 $\displaystyle 4x + k = 3x^2 + 12x + 7$

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

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

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

$\displaystyle 64 - 84 + 12k < 0$

$\displaystyle -20 + 12k < 0$

$\displaystyle 12k < 20$

$\displaystyle k < \frac{5}{3}$.

3. the bit i was looking for an explanation for was- "a straight line has equation $\displaystyle y = 4x + k$ where k is a constant, and a parabola has equation $\displaystyle y = 3x^2 + 12x + 7$ show that the x coordinate of any points of intersection of the line and the parabola satisfy $\displaystyle 3x^2 + 8x + 7 - k = 0$"

4. Yes I did.

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

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

and collecting everything on the one side gives

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

as required.