please help to solve this---**Overall Content:**
The image presents a mathematical problem consisting of a system of two equations and a question asking if the viewer can solve it.
**Textual Information:**
* **Question:** Can You solve ?
* **Equations:**
* X² - Y² = 27
* XY = 18
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答案文本
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Welcome to this mathematical challenge! We have a system of two equations with two unknowns X and Y. The first equation is X squared minus Y squared equals 27, and the second equation is X times Y equals 18. Let's solve this system step by step to find all possible solutions.
Let's start solving this system using the substitution method. From the second equation X Y equals 18, we can express Y in terms of X as Y equals 18 divided by X. Now we substitute this expression for Y into the first equation, giving us X squared minus 18 over X squared equals 27. Simplifying this, we get X squared minus 324 over X squared equals 27.
Now we need to solve this equation. First, multiply both sides by X squared to eliminate the fraction, giving us X to the fourth power minus 324 equals 27 X squared. Rearranging this into standard form, we get X to the fourth power minus 27 X squared minus 324 equals zero. This is a quartic equation, but we can simplify it using substitution. Let Z equal X squared, then our equation becomes Z squared minus 27 Z minus 324 equals zero, which is a quadratic equation.
Now we solve the quadratic equation Z squared minus 27 Z minus 324 equals zero using the quadratic formula. The discriminant is 27 squared plus 4 times 324, which equals 729 plus 1296, giving us 2025. The square root of 2025 is 45. So Z equals 27 plus or minus 45, all divided by 2. This gives us Z equals 36 or Z equals negative 9.
Now we find the X and Y values. Since Z equals X squared, from Z equals 36 we get X equals plus or minus 6. From Z equals negative 9, we get X equals plus or minus 3 i, where i is the imaginary unit. Using Y equals 18 divided by X, we find the corresponding Y values. Our four solutions are: X equals 6, Y equals 3; X equals negative 6, Y equals negative 3; X equals 3 i, Y equals negative 6 i; and X equals negative 3 i, Y equals 6 i. The system has been successfully solved with both real and complex solutions.