Expanding and Solving: (x^2 + 10x + 20)^2 = (x+a)(x+b)(x+c)(x+d)+16
In this article, we will explore the expansion of the quadratic expression (x^2 + 10x + 20)^2
and solve for the values of a
, b
, c
, and d
that satisfy the equation (x+a)(x+b)(x+c)(x+d)+16
.
Expanding the Quadratic Expression
Let's start by expanding the quadratic expression (x^2 + 10x + 20)^2
using the binomial theorem.
$(x^2 + 10x + 20)^2 = x^4 + 20x^3 + 100x^2 + 400x + 400$
Factoring the Expanded Expression
Now, let's factor the expanded expression into the product of four binomials:
$x^4 + 20x^3 + 100x^2 + 400x + 384 = (x+a)(x+b)(x+c)(x+d) + 16$
Equating Coefficients
To find the values of a
, b
, c
, and d
, we can equate the coefficients of the two expressions. Let's start by equating the constant terms:
$384 = abcd + 16$
Subtracting 16 from both sides gives:
$abcd = 368$
Next, let's equate the linear terms:
$400 = a(bcd) + b(acd) + c(abd) + d(abc)$
Now, let's equate the quadratic terms:
$100 = ab(cd) + ac(bd) + ad(bc) + bc(ad) + bd(ac) + cd(ab)$
Finally, let's equate the cubic terms:
$20 = a(bc) + b(ac) + c(ab) + d(ab)$
Solving for a, b, c, and d
Solving the above system of equations, we get:
$a = 2, b = 4, c = 6, d = 8$
Calculating a^2 + b^2 + c^2 + d^2
Now, let's calculate the sum of the squares of a
, b
, c
, and d
:
$a^2 + b^2 + c^2 + d^2 = 2^2 + 4^2 + 6^2 + 8^2 = 4 + 16 + 36 + 64 = \boxed{120}$
Therefore, the final answer is 120.