I've been trying to figure this question out for a while, but I've never come across one like this before. Can anyone help to explain how I would answer this question?
Question: a) express x^2 - 4x + 1 in the form (x + a )^2 + b
b) hence solve the equation x^2 - 4x + 1 = 0 giving your answer in the form p +- sqrt q where p and q are integers
I managed to figure out part a) which was ( x - 2 )^2 - 3
I would expect you to have been given examples like this before being asked the question; but if not, think about adding 3 to both sides, then taking the square root of each side. The nice thing about this form is that x appears in only one place, so you can isolate it more or less the same way you do in a linear equation.
"Question: a) express x^2 - 4x + 1 in the form (x + a )^2 + b"
This is called "completing the square". You should have learned, before you could be expected to do a problem like this, that (x+a)2=x2+2ax+a2.
(You can get that by actually doing the multiplication: (x+a)2=(x+a)(x+a)=x(x+a)+a(x+a)=x2+ax+ax+a2=x2+2ax+a2.)
Compare the first part of that, "x2+2ax", to your "x2−4x". They will be exactly the same if 2a=−4 which means that a=−2. Yes, (x−2)2=x2−4x+4. In order to "complete the square", to make x2−4x a complete square, you need to add 4. Of course you can't just "add 4", that would change the value. But you can do the old trick of "add" and "subtract" the same thing so that you change the "form" but not the value: x2−4x+1=x2−4x+4−4+1=(x2−4x+4)+(−4+1)=(x−2)2−3. That is "in the form (x+ a)^2+ b" with a= -2 and b= -3.
For (b), write x2−4x+1=(x−2)2−3=0. Adding 3 to both sides, (x−2)2=3. Taking the square root of both sides, x−2=±3 so, adding 2 to both sides, x=2±3.
This site uses cookies to help personalise content, tailor your experience and to keep you logged in if you register.
By continuing to use this site, you are consenting to our use of cookies.