Express x^2 - 10x in the form (x+p)^2 + q

[Haroldingo]

Homework Statement

Express x^2 - 10x in the form (x+p)^2 + q

State the value of P and Q

The Attempt at a Solution

I don't know! I don't get it because when I times out the brackets p will always be a number, and there are no numbers that aren't multiplied by x in x squared minus 10 x. Gragh?

 

[Curious3141]

Expand this out: (x+p)^2 + q

Compare the coefficients of this expression to x^2 - 10x (the constant term here is zero, i.e. the expression can be written x^2 - 10x + 0). What equations can you set up to define p and q?

 

[eumyang]

Another way is to complete the square.

 

[Haroldingo]

Ok so I complete the square because I'm more comfortable with that:

x^2 - 10x + 0 = 0

x^2 - 10x = 0

x^2 - 5^2 = 5^2

(x-5)^2 - 25

this would mean p = 5 and q = -25?

This doesn't seem right? Have I gone wrong somewhere?

 

[Gliese123]

Homework Statement

Express x^2 - 10x in the form (x+p)^2 + q

State the value of P and Q

The Attempt at a Solution

I don't know! I don't get it because when I times out the brackets p will always be a number, and there are no numbers that aren't multiplied by x in x squared minus 10 x. Gragh?

Ok so I complete the square because I'm more comfortable with that:

x^2 - 10x + 0 = 0

x^2 - 10x = 0

x^2 - 5^2 = 5^2

(x-5)^2 - 25

this would mean p = 5 and q = -25?

This doesn't seem right? Have I gone wrong somewhere?

It should be right as you wrote:
x²-10x <=> (x+p)(x+p) + q <=> x²+2xp+p²+q.
x²-5x-5x+25-25 <=> x²-10x

p=5, q=-25

 

[evosy1978]

p=-5 q=-25

I find it quicker to half the x term... so there's your p straight away -5

and then sqaure p so -5^2 = 25 and you need +0 so its -25...

another example is
x^2 - 6x + 30

so again half -6 is p= -3
then again square -3 = 9
and i need 30 so 9+21 q =21

 

[Haroldingo]

Cheers all, got the marks! :)

 

[Mark44]

Some quibbles:

Ok so I complete the square because I'm more comfortable with that:
x^2 - 10x + 0 = 0

You're not working with an equation - just an expression.

x^2 - 10x = 0

x^2 - 5^2 = 5^2

Didn't notice earlier, but this isn't correct.
Where did the -10x term go? And how is it that you can add -5^2 to one side of an equation, but add +5^2 to the other. The answer is, you can't do this.

(x-5)^2 - 25

this would mean p = 5 and q = -25?

Starting at the beginning, you have
x² - 10x
= x² - 10x + 25 - 25
= (x - 5)² - 25
= (x + (-5))² + (-25)
I leave it for you to figure out what p and q are.

This doesn't seem right? Have I gone wrong somewhere?

It should be right as you wrote:
x²-10x <=> (x+p)(x+p) + q <=> x²+2xp+p²+q.
x²-5x-5x+25-25 <=> x²-10x

p=5, q=-25

Gliese123, Do not connect expressions with an equivalent sign. Expressions that are equal should be connected with =. Statements such as equations or inequalities can in some cases be connected with the equivalent symbol, <=>.