Use the fact that 25x^2=(5x)^2 to rewrite the given expression as the square of a difference of monomials.
81
Practice makes perfect
A perfect square trinomial is an expression that is the result of squaring either a sum or a difference of monomials.
( a+ b)^2 &= a^2+2 a b+ b^2_()
& Perfect square trinomial
( a- b)^2 &= a^2-2 a b+ b^2_()
Notice that both expression are almost the same. They are different just for the sign in front of the second term. In our case, we are given the following trinomial.
25x^2 - 90x + c
Since the sign of the linear term is negative, we are looking for the square of a difference of monomials. Next, let's rewrite 25 as 5^2 and 90x as 2(45x).
25x^2 - 90x + c = 5^2x^2 - 2(45x) + c
By using the Properties of Exponents we can rewrite the equation above as follows.
25x^2 - 90x + c = (5x_a)^2 - 2(45x)+c
From the above we get that a=5x. Knowing this, we rewrite 45x as 5x* 9.
25x^2 - 90x + c = (5x)^2 - 2* 5x* 9_(a* b) + c
The equation above tells us that b=9 and, since c= b^2, we have that c = 9^2=81.
25x^2 - 90x + c &= (5x)^2 - 2* 5x* 9 + 9^2
&= (5x-9)^2
