# Finding Minimum from a completion of a square

1. Oct 31, 2005

### laker_gurl3

so the question is:

Complete the square of x^2 + 4x - 1 and hence find the position and value of its minimum.

AFter completing the square i have : (x+2)^2 - 5 ,

How do I find the minimum value from that?

2. Oct 31, 2005

### Tide

What is the smallest possible value for $(x+2)^2$?

3. Oct 31, 2005

### laker_gurl3

umm so would x= -2 and y = -5

??

4. Oct 31, 2005

### TD

Indeed

5. Oct 31, 2005

### laker_gurl3

okaay another questoin....
Factorise and hence solve the equation:

2x^3 + 3x^2 - 8x - 12 = 0

I can't seem to remember how to factorize this because there's a cubic expression! how do i start?

6. Oct 31, 2005

### TD

Divisors of the constant term (i.e. -12 here) are possible zeroes of your equation. Try to find such a zero and use the fact that if x = a is a zero, you can factor out (x-a).

7. Oct 31, 2005

### laker_gurl3

okay thanks, i'm going to try that now, by the way is that what you call the remainder theorem?

8. Oct 31, 2005

### laker_gurl3

(x-2)(x+2)(2x+3)

is that right?

9. Oct 31, 2005

### Tide

You can expand that product and see if it produces your original expression.

10. Oct 31, 2005

### laker_gurl3

So there's this question to factorize:

x^4 - 3x^2 - 10 = 0

How do I start this one because I can't find a factor to go into it.

11. Oct 31, 2005

### TD

Would it be easier for you to see if you let $x^2 = t$ so it becomes quadratic?

That would give $$t^2-3t-10=0$$ to factor and then substitute t by x² again.

12. Oct 31, 2005

### laker_gurl3

so the ans is $$\sqrt{5}$$ ?

Last edited: Oct 31, 2005
13. Oct 31, 2005

### Tide

That's one of them! :)

14. Oct 31, 2005

### laker_gurl3

Ohh is it +/- Square root 5?! lol

15. Oct 31, 2005

### TD

Indeed, unless you're working complex since then a quartic equation has 4 solutions. The ones you found are the only real ones though.

16. Oct 31, 2005

### laker_gurl3

help with this last one

Express the function $$\sqrt{3}\sin{2t} - 3\cos{2t}$$ in the form A$$\sin{(2t+\alpha)$$

i have no idea what to do

Last edited: Oct 31, 2005
17. Oct 31, 2005

### Tide

Now you're just guessing!

$$t^2 - 3t - 10 = (t-5)(t+2)$$

18. Oct 31, 2005

### TD

Yes, but t was x².

19. Oct 31, 2005

### laker_gurl3

well if you sub the $$x^2$$ back in to the equation i get $$\pm\sqrt{5}$$. is that not correct? and what of the other question? how should i start what am i suppose to do?

20. Oct 31, 2005

### TD

I would 'expand' sin(2t+a) with the formula for addition of angles, being:

$$\sin \left( {a + b} \right) = \sin a\cos b + \cos a\sin b$$