Bunch of AS level math problems(quadratics,indices etc)

[giddy]

Ok, so I have very little time to nail my AS level math papers and I got stuck at a couple places with a bunch of problems from various chapters. I've collected all my snags and I'm posting them together.
Problem 1
Use the completed square form to factorize 6x² - 5x - 6

The attempt at a solution
(x + 5/2)² - 25/4 - 6
{(x + 5/2)² - (5/2)²} - 6
(by a² - b² = (a-b)(a +b))
[((x + 5/2) - (5/2))((x + 5/2) + (5/2)) ]- 6
[x(x + 10/2)] - 6 >>>>> Which is totally weird because the answer is supposed to be:
(3x +2)(2x-3)

Problem 2
Find the point of intersection of the foll. curves:
x + 2y = 3; x² + xy = 2
I know I need to substitute one equation in the other. I managed all the other problems in the section except this one.
The expected answer is : (1,1) (-4,3.5)

Problem 3
Find values of k such that the straight line y = 2x + k meets the curve with equation x² + 2xy + 2y² = 5 exactly once.
I don't really know what to do, I tried substituting like the previous problem and came up with a complicated mess of an equation with k.

Problem 4
Show that the equation 2^x+1 + 2^x-1=160 can be written in the form 2.5 X 2^x = 160. Find the value of x which satisfies the equation.
I'm a little embarrassed that i do now know how to really tackle this one!?

Problem 5
Solve the inequality : (x-1)(x-6)<0
There something in my book about a table of signs of the factors but I can't make any sense of it!

I'm studying math on my own and I'm a little absent minded sometimes and I miss the obvious when i can even understand complex things, so I'm sorry if any of the questions were silly.

Thanks
Gideon

 

[rock.freak667]

1) First step to make the completed square form is this 6(x²-5x/6)-6.

2) You'd need to show us your working to say if you did something wrong

3) If you substitute for y, you'll get a quadratic in x. If


$$x = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}$$


because it meets the curve only once, means there is only one solution for 'x'. Looking at the formula above. How do you get it so that x is one value and not in the form x=(1±2)/2 or something like that?

4) This will help you a^m*aⁿ=a^m+n. Work backwards with this one.

5) What I'd do is see that the curve crosses the x-axis at 1 and 6. Draw the axis and see for what range of x, the curve lies for less than zero...or below the x-axis as it means.

 

[njama]

Ok, so I have very little time to nail my AS level math papers and I got stuck at a couple places with a bunch of problems from various chapters. I've collected all my snags and I'm posting them together.
Problem 1
Use the completed square form to factorize 6x² - 5x - 6

The attempt at a solution
(x + 5/2)² - 25/4 - 6

This is not correct.

Start by factoring 6.

6(x²-(5/6)x-1)=
=6[(x-5/12)²-25/144-1]=
=6[(x-5/12)²-169/144]

Now use the fact that 169/144 = (c/d)²

Next use a²-b²=(a-b)(a+b)

Problem 2

Hint: for the first equation
2y=x-3
y=(x-3)/2

Now substitute for y in the second equation and solve for x

Problem 5

Ok, if you have a*b<0 what "sign" should the values of a, b have?

Hint: a<0 & b>0 or a>0 & b<0