1. ## disicriment of solutions

$\displaystyle 6x^2+4kx+(k+3)=0$

i)find one solution
ii) two
iii)none

2. Originally Posted by johnsy123
$\displaystyle 6x^2+4kx+(k+3)=0$

i)find one solution
ii) two
iii)none

Quadratic: $\displaystyle ax^2+bx+c=0$

Yours: $\displaystyle 6x^2 + 4kx + (k+3) = 0$

Assuming k is a constant we can match the coefficients:

• $\displaystyle a = 6$
• $\displaystyle b = 4k$
• $\displaystyle c = k+3$

The discriminant is given by: $\displaystyle \Delta = b^2-4ac$

If:
• $\displaystyle \Delta > 0$ then there are two, different, real solutions
• $\displaystyle \Delta = 0$ then there are two, equal, real solutions
• $\displaystyle \Delta < 0$ then there are no real solutions

I will let you figure out the rest of the question

3. Originally Posted by johnsy123
$\displaystyle 6x^2+4kx+(k+3)=0$

i)find one solution
ii) two
iii)none
For all of the above, you need to evaluate the discriminant.

$\displaystyle \Delta = b^2 - 4ac$

$\displaystyle = (4k)^2 - 4(6)(k + 3)$

$\displaystyle = 16k^2 - 24k - 72$.

For one solution, $\displaystyle \Delta = 0$.

So $\displaystyle 16k^2 - 24k - 72 = 0$

$\displaystyle 8(2k^2 - 3k - 9) = 0$

$\displaystyle 8(2k^2 - 6k + 3k - 9) = 0$

$\displaystyle 8[2k(k - 3) + 3(k - 3)] = 0$

$\displaystyle 8(k - 3)(2k + 3)= 0$.

So $\displaystyle k - 3 = 0$ or $\displaystyle 2k + 3 = 0$

$\displaystyle k = 3$ or $\displaystyle k = -\frac{3}{2}$, for one solution.

No solutions: $\displaystyle \Delta < 0$.

So $\displaystyle 16k^2 - 24k - 72 < 0$.

To evaluate this, you will need to complete the square.

$\displaystyle 16\left(k^2 - \frac{3}{2}k - \frac{9}{2}\right) < 0$

$\displaystyle 16\left[k^2 - \frac{3}{2}k + \left(-\frac{3}{4}\right)^2 - \left(-\frac{3}{4}\right)^2 - \frac{9}{2}\right] < 0$

$\displaystyle 16\left[\left(k - \frac{3}{4}\right)^2 - \frac{9}{16} - \frac{72}{16}\right] < 0$

$\displaystyle 16\left[\left(k - \frac{3}{4}\right)^2 - \frac{81}{16}\right] < 0$

$\displaystyle \left(k - \frac{3}{4}\right)^2 - \frac{81}{16} < 0$

$\displaystyle \left(k - \frac{3}{4}\right)^2 < \frac{81}{16}$

$\displaystyle \sqrt{\left(k - \frac{3}{4}\right)^2} < \sqrt{\frac{81}{16}}$

$\displaystyle \left|k - \frac{3}{4}\right| < \frac{9}{4}$

$\displaystyle -\frac{9}{4} < k - \frac{3}{4} < \frac{9}{4}$

$\displaystyle -\frac{3}{2} < k < 3$.

2 solutions: $\displaystyle \Delta > 0$.

So $\displaystyle 16k^2 - 24k - 72 > 0$.

To evaluate this, again you will need to complete the square.

$\displaystyle 16\left(k^2 - \frac{3}{2}k - \frac{9}{2}\right) > 0$

$\displaystyle 16\left[k^2 - \frac{3}{2}k + \left(-\frac{3}{4}\right)^2 - \left(-\frac{3}{4}\right)^2 - \frac{9}{2}\right] > 0$

$\displaystyle 16\left[\left(k - \frac{3}{4}\right)^2 - \frac{9}{16} - \frac{72}{16}\right] > 0$

$\displaystyle 16\left[\left(k - \frac{3}{4}\right)^2 - \frac{81}{16}\right] > 0$

$\displaystyle \left(k - \frac{3}{4}\right)^2 - \frac{81}{16} > 0$

$\displaystyle \left(k - \frac{3}{4}\right)^2 > \frac{81}{16}$

$\displaystyle \sqrt{\left(k - \frac{3}{4}\right)^2} > \sqrt{\frac{81}{16}}$

$\displaystyle \left|k - \frac{3}{4}\right| > \frac{9}{4}$

$\displaystyle k - \frac{3}{4} < -\frac{9}{4}$ or $\displaystyle k - \frac{3}{4} > \frac{9}{4}$

$\displaystyle k < -\frac{3}{2}$ or $\displaystyle k > 3$.