Quadratics Q: The curve C has equation y = x^2 - 4ax + 3a^2?

Question

I'm having difficulties with the following question:

The curve C has equation y = x^2 - 4ax + 3a^2, where a is a positive constant.

• Factorise the expression for y, and hence sketch the curve, showing clearly its points of intersection with the coordinate axes

• Write down the solution of the inequality y = x^2 - 4ax + 3a^2 < 0

I know that the factorised version is (x-a)(x-3a) but I was told that without an explanation of how to get it.

I'd be very grateful if somebody please help me on how to answer.
Thanks in advance,
MAK

Answer 1

if you have a quadratic of the form:

y = x^2 + px + q (where p and q are constants)

If it can be factorsed then it can turn into the form

y = (x + r)(x+t)

So that when you multiply everything together you get the above equation back
y = x^2 + (r+t)x + r*t

so that

p = r + t
and
q = r * t

You then look for values of r and t which satisfy this condition (positive or negative values)

in your case (r= -a) (t = -3a) (p = -4a) and (q = 3a^2)

the intersections are where y=0, so look for values of x which make (x-a)(x-3a) equal to zero. There should only be two and it should be obvious if you look at the values which make each bracket equal to zero.

Answer 2

u have to use the quadratic equations to get what the 2 factors are by jus treating "a" as a constant. ONce you have the 2 factors, x minus each of those factors mutliplied by each other equals 0 and so that's how it was factored. Find what the solutions/zeros are via quadratic equation and then put it in factored form.

Answer 3

P(at^2,2at) 'if its a factor of the curve, once you replace the x and y with the factor coordinates it would be g=g x=at^2 y=2at y^2=4ax (2at)^2 = 4aat^2 => 4a^2t^2 = 4a^2t^2 so the factor lies on the curve

Answer 4

1.)Y=(X-a)(X-3a)
y=0 =>x=a or x=3a then the points (a,0) & (3a,0)is of x-axis, and (0,3a^2) is of y-axis.
2.) y<0 <=> (X-a)(X-3a)<0, then x is in (a,3a).The solution is (a,3a).end!

Answer 5

(x^2 - ax) +(3a^2 - 3ax)= (x-a)x + 3a(a-x) =(x-a)(x-3a)

x^2 - 4ax +4a^2 - a^2 = (x-2a)^2 - a^2 <0

Answer 6

zero!