# Roots question

1. Jul 4, 2008

### crays

Show that the equation x2 + (3$$\alpha$$ - 2)x + $$\alpha$$($$\alpha$$ - 1) = 0 has real roots for all values of α $$\in$$ IR.

How do i do this? Do i just use the
b2 + 4ac $$\geq$$ 0 ?

2. Jul 4, 2008

### dirk_mec1

Yes that should be correct.

3. Jul 4, 2008

But i can't factorize it out. i get 5$$\alpha[tex]2 - 2[tex]\alpha[tex] + 4 [tex] \geq[tex] 0 Do i use the formula? and write [tex]\alpha[tex] ranges that will satisfy it? here's another question, show that x2 - x + 1 has the same sign for all values of x. How do i show? Just x2 - x [tex]\geq[tex] 0 will do ? 4. Jul 4, 2008 ### malawi_glenn you can show that by drawing the graph for the function f(x) = x^2 - x + 1 and check that the function never crosses the x-axis. 5. Jul 4, 2008 ### crays How do i draw a graph without knowing its roots? 6. Jul 4, 2008 ### malawi_glenn so you don't know how to draw graphs using differential calculus? otherwise, just find the roots to x^2 - x + 1 , (it is very simple) - if you want. 7. Jul 4, 2008 ### crays Sorry for the late reply. Yes i don't know how to do so. But X^2 + x - 1, i use the formula to find the roots? then just draw the graph? 8. Jul 4, 2008 ### rock.freak667 [tex]x^2-x+1$$
you could just use the disciminant or find the minimum point or putting it in the completing the square form.

9. Jul 4, 2008

### Defennder

Which question are you doing now? For the first one, just apply the discriminant condition and you'll get a quadratic equation in terms of $\alpha$. From there, you should use the discriminant condition again for the alpha quadratic to see if the quadratic equation for alpha can ever be negative.

The second question is answered much the same. You have x^2 - x +1. What condition must the discriminant satisfy such that f(x) does not cross the x-axis? Check also that the graph is above the x-axis to begin with.

10. Jul 4, 2008

### crays

so do i just write x^2 - x >= 0 ? In that condition, it will never touches. I still don't get how should i draw a graph. If i use completing the square, i found (x - 1/2)^2 + 1 + 1/4

ps. I got it already, thanks guys.

11. Jul 4, 2008

### Defennder

Where did x^2 - x come from? The discriminant is b^2 - 4ac. There should be no x in the discriminant, only numbers. As for graph sketching, note that the graph is either a u-shape or n-shape depending on the sign of the coefficient of the x^2 term. That should help you visualise if the f(x) is always positive or negative.

12. Jul 4, 2008

### crays

Erm with the discriminant i found -3. So i write -3 < 0 ?

13. Jul 5, 2008

### Defennder

What does that tell you about f(x)? More specifically what does that tell you about how the graph of f(x) should look like?

14. Jul 5, 2008

### crays

it means the value is below -3? the graph should looks like a U right ?

15. Jul 5, 2008

### Defennder

Yes and you also need to add in what the discriminant tells you.

16. Jul 5, 2008

### crays

thanks. Mind if i ask another question?
Determine the condition to be satisfied by k such that the expression 2x^2 + 6x + 1 + k(x^2 + 2) is positive for all x .
Do i use discriminant for this?

17. Jul 5, 2008

### dirk_mec1

Note that

$$2x^2>0$$
$$kx^2>0,\ k \geq 0$$

and so on. You could also use the derative.

18. Jul 5, 2008

### Defennder

Yes you can discriminant for this. I assume the reason why you didn't use calculus is because you haven't learnt that yet.

19. Jul 5, 2008

### crays

erm, is differentiation and integration called calculus?
I tried using the discriminant for it. But the value is off too.

20. Jul 5, 2008

### Defennder

I don't know what you mean by 'off'. Can you post your working here?