1. ## Solving Equations

Hi everyone,

I was working on some solving equations in preparation for university, and came across a few that I just could not understand how to do them. I have the answers, but do not know how to get to them, how painful....

1. Find a cubic function of x whose graph has x-intercepts equal to -1 and 2, and in which f(0)=6 and f(3)=12.

2. The roots of the equation x^2+px+1=0 are a and b, and the roots of the equation x^2-9x+q=0 are a+2b and 2a+b. (a and b are greek symbols).

3. If the roots of the equation x^3-15x^2+cx-105, are a, a+b, and a-b, determine the values of a,b, and c.

2. 3. $x^3 - 15x^2 + cx - 105$ has roots $x = a, x = a+ b, x = a - b$.

That means $f(a) = 0, f(a + b) = 0, f(a - b) = 0$.

So $a^3 - 15a^2 + ac - 105 = 0$
$(a+b)^3 - 15(a+b)^2 + (a+b)c - 105 = 0$
$(a-b)^3 - 15(a-b)^2 + (a-b)c - 105 = 0$.

Solve these three equations simultaneously for $a,b,c$.

3. Originally Posted by KelvinScale
Hi everyone,

I was working on some solving equations in preparation for university, and came across a few that I just could not understand how to do them. I have the answers, but do not know how to get to them, how painful....

1. Find a cubic function of x whose graph has x-intercepts equal to -1 and 2, and in which f(0)=6 and f(3)=12.

[snip]
Use the model $f(x) = a(x + 1)(x - 2)(x - b)$ (why?) and use the fact that f(0) = 6 and f(3) = 12 to get two simuatenous equations for a and b.

Originally Posted by KelvinScale
[snip]
2. The roots of the equation x^2+px+1=0 are a and b, and the roots of the equation x^2-9x+q=0 are a+2b and 2a+b. (a and b are greek symbols).

[snip]
$(x - a)(x - b) = x^2 + px + 1$. Therefore:

$ab = 1$ .... (1)

$a + b = -p$ .... (2)

In a similar way, $(x - a - 2b) (x - 2a - b) = x^2 - 9x + q$. Therefore:

...... (3)

....... (4)

Solve equations (1), (2), (3) and (4) simultaneously to get p and q.

If you need more help, please show all your work and say where you get stuck.