Thread: Using the determinant to find real roots

1. Using the determinant to find real roots

Find the range of values of k that give this equation two distinct real roots. 3x^2 +kx +2=0.
I have used the determinant and get the answer k >+/- 2X6^1/2. The answer book says k< -2X6^1/2 and k>2X6^1/2. Can someone explain why this is. I am trying to teach myself A level maths (1st year)

2. First of all I think that you are refering to the discriminant:

$\displaystyle \begin{gathered} \Delta = k^2 - 24 > 0 \hfill \\ \Leftrightarrow k^2 > 24 \hfill \\ \Leftrightarrow k > 2\sqrt 6 \quad or\quad k < - 2\sqrt 6 \hfill \\ \end{gathered}$

3. Discriminant, not determinant

Sounds like you need to use the discriminant, not the determinant. Determinants are usually associated with matrices, and discriminants with quadratic equations and the quadratic formula.

The discriminant for a quadratic equations is given by b^2-4ac. Note that this is the part of the quadratic equation that is under the square root. There are three cases for the discriminant:

1) If it is positive, you will get two distinct, real roots (one from the positive value, one from the negative value of the square root of the discriminant).

2) If it is zero, you will get one real root (since -b +/- 0 is just -b; the +/- doesn't really do anything, since you're adding/subtracting zero)

3) If it is less than zero, you will get no real roots (since you'll be taking the square root of a negative number).

So, you need to find all values for k that makes the discriminant positive. Your discriminant is k^2 - (4)(3)(2), or k^2 - 24. Make an inequality with this greater than zero.

This means k^2 > 24, which means either k > sqrt(24) or k < -sqrt(24) (since squaring it will remove the negative sign). Simplify these inequalities and you should be in good shape!

Todd Werner
Director, Mathnasium West LA
Math Tutoring at Mathnasium Math Learning Centers

4. Hello, newbold!

Find the range of values of $\displaystyle k$ that give this equation
two distinct real roots: .$\displaystyle 3x^2 +kx +2\:=\:0$
You work is correct ... up to a point.

The discriminant is: .$\displaystyle k^2 - 24$ .which must be positive.

. . So we have: .$\displaystyle k^2 - 24 \:>\:0\quad\Rightarrow\quad k^2 \:>\:24$

Now, be very very careful . . .

The next step gives us: .$\displaystyle |k| \:>\:\sqrt{24}$

. . which means: .$\displaystyle k < -\sqrt{24}\:\text{ or }\:k \:> \:\sqrt{24}$

If this is not evident, test some values on the number line.
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determinant formula for root

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