Question

x²+(x+5)²=625,Find the roots of the given equation by the method of perfect square.

Answer 1

x²+x²+25+10x=625

2x²+25+10x = 625

2x²+10x= 600

2x²+10x-600 =0

x²+5x-300=0

x²+20x-15x-300=0

x(x+20)-15(x+20)=0

x+20=0 ; x-15=0

x=-20 ; x=15

Answer 2

We will rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

 x^2+(x+5)^2-(625)=0

Pull out like factors :

2x2 + 10x - 600  =   2 • (x2 + 5x - 300)

Factoring  x2 + 5x - 300

Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -15  and  20  

                    x2 - 15x + 20x - 300

Add up the first 2 terms, pulling out like factors :

                   x • (x-15)

             Add up the last 2 terms, pulling out common factors :

                   20 • (x-15)

Add up the four terms of step 4 :

                   (x+20)  •  (x-15)

            Which is the desired factorization

2 • (x + 20) • (x - 15)  = 0

 x  =  ( -5 ± 35) / 2

x =(-5+√1225)/2=(-5+35)/2= 15