Management Notes

Management Notes

Reference Notes for Management

What are the solutions of the equation x4 + 95×2 – 500 = 0? use factoring to solve.

What are the solutions of the equation x4 + 95×2 – 500 = 0? use factoring to solve.

1. x=+- sqrt 5 and x = ±10
2. x=+- sqrt i5 and x = ±10i
3. x=+- sqrt 5 and x = ±10i
4. x=+- sqrt i5 and x = ±10

d. x=+- sqrt i5 and x = ±10

Let’s break down the solution to the equation using factoring, and then explain why the correct answer is x=± sqrt5 and x=±10.

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

Step 1: Substitution Let’s make a substitution to simplify the equation. Consider, . Therefore, the equation becomes:

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

So, the solutions for are and .

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

For

For :

Therefore, the correct solutions for and .

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

a. and

This choice incorrectly lists as a solution instead of . 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 correctly but misses the complex root .

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. 5 and

This option inaccurately includes as a solution instead of the correct real root . It correctly identifies the complex root , but it substitutes the imaginary unit  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. and

This option correctly identifies the real root sqrt but lists as the complex solution instead of . 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.

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