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

##
#2x^2 + 3kx +k=0#

##### 2 Answers

#### Explanation:

#"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"#

#"here "Delta>0" is required"#

#2x^2+3kx+k=0larrcolor(blue)"is in standard form"#

#"with "a=2,b=3k" and "c=k#

#rarrDelta=(3k)^2-(4xx2xxk)=9k^2-8k#

#"rArr9k^2-8k>0#

#"the left side is a quadratic with positive leading"#

#"coefficient and zeros at "k=0" and "k=8/9#

graph{9x^2-8x [-10, 10, -5, 5]}

#"Thus it is positive when "k<0" or "k>8/9#

#k in(-oo,0)uu(8/9,oo)#

#### Explanation:

is a quadratic equation

and to find the roots of

we use the following formula

so in the quadratic equation

If **the discriminant**

**two real solutions** .

**no real solutions** .

**one real solution** .

**Substitute in the discriminant**

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