How do you factor p2 - 60p + 900 = 0?

Question

the answer shows (p - 30)2 = 0 (the two is supposed to be mean squared)

but i don't understand how to get to that point

Answer 1

p^2 - 60p + 900 = 0 gets factored to
(p + x)(p + y) = 0
You need to find x & y so that when you multiply them you get 900 (the last number), and when you add them you get -60 (the middle number).

Since 900 is a positive number, both x & y need to be positive or they both need to be negative. Since -60 is a negative number, we can conclude that both x & y need to be negative. When you factor 900, you find that these combinations give you a product of 900:
-1 x -900
-2 x -450
-3 x -300
-4 x -225
-5 x -180
-6 x -150
-9 x -100
-10 x -90
-12 x -75
-15 x -60
-18 x -50
-20 x -45
-25 x -36
-30 x -30

Since you need to add the two numbers to get -60, the only one that satisfy this is the last option: -30 x -30.

So x = -30 and y = -30 and you get:
(p - 30)(p - 30) = 0 which also equals
(p - 30)^2 = 0

Answer 2

The long way (which should always work if there is an answer) is to use the formula to solve quadratic equations:
('quadratic equation' simply means, a polynomial of degree 2)

write the equation in the following form (yours already is)

a x^2 + b x + c = 0
(here, a = 1, b = -60, c = 900).

Then the roots are at:

[ -b +/- SQRT( b^2 - 4ac)] / 2a

In this case, b^2 - 4ac = 3600 - 4*900 = 0
Therefore, the square root disappears (it is zero) and you are left with
-b / 2a = 60/2 = 30

The equation is true when p = 30 (try it out).

If that is true, then the left side should be divisible by (p-30). If you do the division, you find that the other factor is (p-30) as well (which was already hinted at by our square root being zero).

so (p-30)(p-30)= p^2 -60p + 900
and, when p = 30, we get :
0*0 = 900 -60(30) + 900 = 900 - 1800 + 900 = 0

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Normally, most quadratics that have solutions do have two distinct roots and you get two factors of this type:

(p - x)(p - y)

where x and y are called 'roots' of the equation.

In your example, you have a 'double root', also called a 'repeated root', or a second degree root. Some will even say: you have two roots which happen to be identical.
All different ways of saying the same thing.

A quadratic equation can never have more than 2 roots.

In complex numbers, you will see that quadratic equations always have exactly two roots (and, in some cases like in your problem, both can be the same).

Answer 3

when you factor a trinomial, you want to think about what numbers multiply to give you 900, and add to give you -60....Well, if you think about it 900 is a perfect square: 30 X 30

if you add 30+30 you will get 60....so we know 30 and 30 have to be our numbers. In order to get a -60, the 30's must both be negative (and that makes sense since -30 X -30 is positive 900)

So we have: (p - 30)(p - 30)

Since both factors are the same, we can shorten it by writing (p-30)^2

Hope that helped!! =)

Answer 4

(p-30)^2 = p^2-2*p*30 +30^2 = p^2-60p+900