Answer:
x = 12
Explanation:
First, we are going to expand that squared binomial. I like to use the FOIL method, standing for firsts, outsides, insides, lasts and representing what terms are multiplied together in order to expand.
(x + 9)² = (x + 9)(x + 9)
Firsts: x(x) = x²
Outsides: x(9) = 9x
Insides: 9(x) = 9x
Lasts: 9(9) = 81
Expanded, this square binomial is: x² + 9x + 9x + 81
Combine like terms: x² + 18x + 81
Back to the original equation, we can now substitute (x + 9)² and combine like terms again.
x² + 18x + 81 - 441 = 0
x² + 18x - 360 = 0
Now, lets factor this trinomial. To factor a trinomial in ax² + bx + c form, we find two factors of c whose sum is equal to b. So, what two numbers when multiplied equal -360 but are added together to make 18? These numbers are 12 and -30. So let's expand the equation again and factor it once more.
x² - 12x + 30x - 360 = 0
Now, we can factor pairs of terms
(x² - 12x) + (30x - 360) = 0
x(x - 12) + 30(x - 12) = 0
So (x - 12)(x + 30) = 0 is our new equation. To solve for x, set each of these binomials equal to zero.
x - 12 = 0 x + 30 = 0
x = 12 x = -30
If we substitute x into the original length of each side of the square we get measurements of -21 and 31 (-30 + 9 and 12 + 9, respectively). Because length as a distance cannot be negative, the value of x cannot be the number that causes a negative answer, thus. x = -30 is out.
This leaves us with our answer, x = 12.