[sp]$a$ and $b$ satisfy the equation $(x^2-4)^2 = 5-x$, or $x^4 - 8x^2 + x + 11 = 0\quad(*).$anemone said:Let $a, b, c, d$ be real numbers such that \(\displaystyle a=\sqrt{4-\sqrt{5-a}}\), \(\displaystyle b=\sqrt{4+\sqrt{5-b}}\), \(\displaystyle c=\sqrt{4-\sqrt{5+c}}\) and \(\displaystyle d=\sqrt{4+\sqrt{5+d}}\). Calculate $abcd$.
Opalg said:[sp]$a$ and $b$ satisfy the equation $(x^2-4)^2 = 5-x$, or $x^4 - 8x^2 + x + 11 = 0\quad(*).$
$c$ and $d$ satisfy the equation $(x^2-4)^2 = 5+x$, or $x^4 - 8x^2 - x + 11 = 0\quad(**).$
But if $x$ satisfies (**) then $-x$ satisfies (*). So the roots of (*) are $a,b,-c,-d.$ Thus $abcd$ is the product of the roots of (*), namely 11.[/sp]
The formula for solving for $abcd$ given real numbers is simply $abcd = a \times b \times c \times d$, where $a$, $b$, $c$, and $d$ are real numbers.
The steps for solving for $abcd$ given real numbers are as follows:
1. Write out the formula $abcd = a \times b \times c \times d$
2. Identify the values of $a$, $b$, $c$, and $d$
3. Substitute the values into the formula
4. Perform the necessary operations (multiplication)
5. Simplify the expression to get the final value of $abcd$.
Yes, you can still solve for $abcd$ even if one or more of the numbers is a fraction. Simply follow the same steps as you would for solving with whole numbers.
Solving for $abcd$ given real numbers helps us to understand the relationship between the numbers and how they interact with each other in a multiplication operation. It also allows us to find the value of $abcd$ in various real-life scenarios, such as in physics equations or financial calculations.
Yes, there are alternative methods for solving for $abcd$ given real numbers, such as using a calculator or creating a table to organize the values and their corresponding products. However, the formula $abcd = a \times b \times c \times d$ is the most straightforward and efficient method for solving this problem.