(x^2 + 10x + 20)^2 = (x+a)(x+b)(x+c)(x+d)+16 จงหา A^2+b^2+c^2+d^2

Suzhoumeite

3 min read

·

Jul 03, 2024

65

Share

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.