1. ## discriminant

hi, i just want to check my answer against someone else's here, my textbook tells me a different answer to what i'm getting. i think the books wrong, could someone tell me if i'm right please:

by considering the discrimination of the resulting quadratic, determine the number of points of intersection of the following line and curve:

line $\displaystyle 3x - 2y = 5$ and curve $\displaystyle y = 3x^2 - 4x - 2$

the book says two points of intersection and discriminant = 97
but i say two points of intersection and discriminant = 241

thanks

2. Originally Posted by mark
hi, i just want to check my answer against someone else's here, my textbook tells me a different answer to what i'm getting. i think the books wrong, could someone tell me if i'm right please:

by considering the discrimination of the resulting quadratic, determine the number of points of intersection of the following line and curve:

line $\displaystyle 3x - 2y = 5$ and curve $\displaystyle y = 3x^2 - 4x - 2$

the book says two points of intersection and discriminant = 97
but i say two points of intersection and discriminant = 241

thanks
Are you sure the answer given wasn't $\displaystyle \frac{97}{4}$?

They will intersect where the functions are equal.

$\displaystyle y = \frac{3}{2}x - \frac{5}{2}$

$\displaystyle y = 3x^2 - 4x - 2$.

Thus

$\displaystyle \frac{3}{2}x - \frac{5}{2} = 3x^2 - 4x - 2$

$\displaystyle 0 = 3x^2 - \frac{11}{2}x + \frac{1}{2}$.

The discriminant

$\displaystyle \Delta = \left(-\frac{11}{2}\right)^2 - 4\left(3\right)\left(\frac{1}{2}\right)$

$\displaystyle = \frac{121}{4} - 6$

$\displaystyle = \frac{97}{4}$.

So the discriminant is $\displaystyle \frac{97}{4}$ so there will be two points of intersection.

3. no it just says 97, you've given a very complicatied way of figuring it out compared to the books method

4. Originally Posted by mark
no it just says 97, you've given a very complicatied way of figuring it out compared to the books method
I don't see what's so complicated...

Let's try it a different way

Substitute $\displaystyle y = 3x^2 - 4x - 2$ into the first equation.

$\displaystyle 3x - 2(3x^2 - 4x - 2) = 5$

$\displaystyle 3x - 6x^2 + 8x + 4 = 5$

$\displaystyle 0 = 6x^2 - 11x + 1$

Checking the discriminant

$\displaystyle \Delta = (-11)^2 - 4(6)(1)$

$\displaystyle = 121 - 24$

$\displaystyle = 97$.

So the answer is two solutions with $\displaystyle \Delta = 97$.

Having said that, my first method was also correct, because notice that if you divide both sides of the equation

$\displaystyle 0 = 6x^2 - 11x + 1$

by 2, we get the same equation as before

$\displaystyle 0 = 3x^2 - \frac{11}{2} + \frac{1}{2}$.

5. you're right, the method i used was a bit ridiculous now i look at it, probably why i got it wrong. thanks though