Find x when P = 21. P = x(x - 1) / 2

Given the equation:

P = x(x - 1)/2

Determine the value of x, if P = 21.

Solution:

Equate the value of P to the equation, then solve for x.

21 = x ( x - 1) /2

21 =( x^2 - x)/2

Multiply both sides by 2 in order to cancel out 2 in the denominator.

(2)(21) = (2)(x^2 - x) /2

x^2 - x = 42

Transpose -42 to the other side of the equation, hence it must be taken to note that in transposing a number, the sign changes.

= x^2 - x - 42

a =1

b = -1 

c= -42

In order to solve for the roots or factors of the equation, we simply must look at the 2nd and 3rd value. First we must think of two numbers that when added, the answer is -1, and when multiplied, the answer is -42. Hence, if we are unable to find the number, we will use the quadratic formula.

The factors that satisfy the equation are

( x + 6 ) ( x - 7 )

To determine if the factors are true, we simply needed to do the FOIL Method. 

Multiply the first term of the first equation, to the first and last term of the second equation. Then, multiply the last term of the first equation, to the first and last term of the second equation.

To clearly see how it works, here's a step by step solution.

= (x)(x)
=x^2
First term of the first equation multiplied to the first term of the second equation.

=(x)(-7)
= -7x
First term of the first equation multiplied to the last term of the second equation.

=(6)(x)
=6x
Last term of the first equation multiplied to the first term of the second equation.

=(6)(-7)
= -42
Last term of the first equation multiplied to the fast term of the second equation.

= x^2 - 7x + 6x - 42

= x^2 - x - 42

Now that we've determine that the factors are true, we simply needed to equate the factors to 0 and solve for x.

x + 6 = 0

x - 7 = 0

Transpose 6 and  -7 to the other side of the equation, hence it must be taken to note that in transposing a number, the sign changes.

x = -6

x = 7

Final answer:
x = -6

x = 7