Let’s break down the solution to the equation x4+95x2−500=0 using factoring, and then explain why the correct answer is x=± sqrt5 x=±i5​ and x$x=\pm 10$.

Correct Solution Explanation: d. x=+- sqrt i5 and x = ±10

Step 1: Substitution Let’s make a substitution to simplify the equation. Consider, y=x2. Therefore, the equation becomes: y2+95y−500=0

Step 2: Factoring Now, let’s factorize the quadratic equation y2+95y−500=0 to solve for $y$. The equation factors to $(y+100)(y-5)=0$.

So, the solutions for $y$ are $y=-100$ and $y=5$.

Step 3: Re-substitution Now, recall that we substituted y=x2.

For $y=-100$: 

x2=−100

 x=±−100​  

$x=\pm 10i$

For $y=5$:

x2=5

x=±5​

Therefore, the correct solutions for x=± sqrt 5​ and $x=\pm 10i$.

Explanation for Incorrect Answers:

Now, let’s analyze why the other options provided are incorrect:

a. x=±5​ and $x=\pm 10$

This choice incorrectly lists $x=\pm 10$ as a solution instead of $x=\pm 10i$. The mistake lies in the misinterpretation of complex roots. When solving a quartic equation, both real and complex roots should be considered. However, this option only considers the real root sqrt 5​ correctly but misses the complex root $10i$.

The error in this option results from a misunderstanding of complex numbers’ relationship to equations and disregarding the existence of complex solutions when dealing with higher-order polynomial equations.

b. 5x=±i5​ and $x=\pm 10i$

This option inaccurately includes x=±i5​ as a solution instead of the correct real root x=±5​. It correctly identifies the complex root $x=\pm 10i$, but it substitutes the imaginary unit $i$ into the square root where it should not be present.

This mistake stems from confusing the representation of imaginary numbers within the square root with the actual value of the square root of 5.

The misunderstanding here is regarding the distinction between representing the square root of a complex number and incorrectly presenting an imaginary component within the square root itself.

c. x=±5​ and $x=\pm 10i$

This option correctly identifies the real root sqrt 5​ but lists $x=\pm 10i$ as the complex solution instead of $x=\pm 10$. The mistake here lies in presenting the incorrect complex roots. This indicates a misinterpretation of the roots obtained from the quadratic equation in connection to the original quartic equation.

The error occurs due to confusion between the real and imaginary components, mislabeling the nature of the roots as either purely imaginary or real, when in reality, both real and complex roots are present in the solutions.

In summary, the incorrect options stem from various misconceptions and misinterpretations related to solving quartic equations, specifically in identifying both real and complex roots. Mistakes include misrepresentation of imaginary units within square roots, mislabeling real and imaginary components, and neglecting the existence of complex roots.

Understanding the distinct nature of real and complex roots, correct substitution of values, and proper handling of complex numbers in equations is crucial for arriving at accurate solutions for quartic equations and similar higher-order polynomial equations.