# What are the set of values for which this equation has real distinct roots?

## $2 {x}^{2} + 3 k x + k = 0$

Apr 18, 2018

$k < 0 \text{ or } k > \frac{8}{9}$

#### Explanation:

$\text{to determine the nature of the roots use the "color(blue)"discriminant}$

•color(white)(x)Delta=b^2-4ac

• " If "Delta>0" then real distinct roots"

• " If "Delta=0" then real and equal roots"

• " If "Delta<0" then complex roots"

$\text{here "Delta>0" is required}$

$2 {x}^{2} + 3 k x + k = 0 \leftarrow \textcolor{b l u e}{\text{is in standard form}}$

$\text{with "a=2,b=3k" and } c = k$

$\rightarrow \Delta = {\left(3 k\right)}^{2} - \left(4 \times 2 \times k\right) = 9 {k}^{2} - 8 k$

"rArr9k^2-8k>0

$\text{the left side is a quadratic with positive leading}$
$\text{coefficient and zeros at "k=0" and } k = \frac{8}{9}$
graph{9x^2-8x [-10, 10, -5, 5]}

$\text{Thus it is positive when "k<0" or } k > \frac{8}{9}$

$k \in \left(- \infty , 0\right) \cup \left(\frac{8}{9} , \infty\right)$

Apr 18, 2018

$k < 0$$\text{ , }$$k > \frac{8}{9}$

#### Explanation:

$2 {x}^{2} + 3 k x + k = 0$
is a quadratic equation

and to find the roots of $x$ of quadratic equations

$\textcolor{g r e e n}{\text{Example :}}$$\textcolor{b l u e}{a {x}^{2} + b x + c = 0}$

we use the following formula

color(blue)(x=(-b+-sqrt(b^2-4ac))/(2a)
color(blue)("where the term "(b^2-4ac) " is the discriminant"

so in the quadratic equation

If the discriminant
${b}^{2} - 4 a c > 0$$\rightarrow$the equation has two real solutions .
${b}^{2} - 4 a c < 0$$\rightarrow$the equation has no real solutions .
${b}^{2} - 4 a c = 0$$\rightarrow$the equation has one real solution .

$2 {x}^{2} + 3 k x + k = 0$

$a = 2$$\text{ , }$$b = + 3 k$$\text{ , }$$c = k$

Substitute in the discriminant

$9 {k}^{2} - \left(4\right) \left(2\right) \left(k\right)$

so in order to get the real distinct roots of the function

color(blue)("discriminant">0

$9 {k}^{2} - 8 k > 0$

$k \left(9 k - 8\right) > 0$

$k < 0$$\text{ , }$$k > \frac{8}{9}$