# Find a and m in this equation: x^2+2(m-a)x+3am-2=0

#### Vali

##### New member
[FONT=&quot]There is the following equation: [/FONT]
[FONT=&quot]x^2+2(m-a)x+3am-2=0 | a,m ∈ R[/FONT]
[FONT=&quot]a) Find a such that the euation has real roots, ∀ m ∈ R.[/FONT]
[FONT=&quot]b) Find m such that the euation has real roots, ∀ a ∈ R.[/FONT]
[FONT=&quot]I put the condition: DELTA >= 0[/FONT]
[FONT=&quot]I found DELTA but since this step I have problems..[/FONT]
[FONT=&quot]Sorry for my english!
The answer should be: |a| <= sqrt(8/21) ; |m| <= sqrt(8/21)
My try:

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#### Vali

##### New member
idea

Can anyone give me a little help or an idea?

#### stapel

##### Super Moderator
Staff member
There is the following equation:
x^2+2(m-a)x+3am-2=0 | a,m ∈ R
a) Find a such that the equation has real roots, ∀ m ∈ R.
b) Find m such that the equation has real roots, ∀ a ∈ R.
I put the condition: DELTA >= 0

For what does "DELTA" stand? How does it relate to the posted question?

I found DELTA but since this step I have problems.
What were your steps? What did you obtain? How did this relate to "a" and "m"?

The answer should be: |a| <= sqrt(8/21) ; |m| <= sqrt(8/21)
My try:
The image in the attachment appears to show the following:

$$\displaystyle \Delta\, =\, h\, (a^2\, -\, 5am\, +\, m^2\, +\, 2)$$

$$\displaystyle a^2\, -\, 5am\, +\, m^2\, +\, 2\, \geq\, 0$$

$$\displaystyle \Delta '\, \geq\, 0$$

$$\displaystyle \Delta '\, =\, 25m^2\, -\, 3m^2\, -\, 8\, =\, )$$

$$\displaystyle \Delta '\, =\, 21m^2\, -\, 8$$

$$\displaystyle 21m^2\, -\, 8\, \geq\, 0$$

$$\displaystyle 21m^2\, \geq\, 8\, =\, )$$

$$\displaystyle \big|m\big|\, \geq\, \sqrt{\dfrac{8}{21}\,}$$

Please reply with confirmation or corrections, along with a clear explanation of what you're doing. Thank you! 