Problem 16 - A Graphical Approach to Precalculus with Limits

Question

Find all complex solutions of each equation. Do not use a calculator. $$x^{4}-11 x^{2}+10=0$$

Step-by-Step Solution

Answer

The complex solutions are \( x = \pm 1, \pm \sqrt{10} \).

Step 1: Recognize the Equation Type

The given equation is a quartic (fourth-degree) polynomial: \( x^4 - 11x^2 + 10 = 0 \). This equation can be simplified by recognizing it as a quadratic in terms of \( y = x^2 \). Thus, transform it into \( y^2 - 11y + 10 = 0 \).

Step 2: Factor the Quadratic Equation

Factor the quadratic equation \( y^2 - 11y + 10 = 0 \). Look for two numbers that multiply to 10 and add to -11. These numbers are -1 and -10. So, the equation factors as: \((y - 1)(y - 10) = 0\).

Step 3: Solve for \( y \)

Set each factor in the equation \( (y - 1)(y - 10) = 0 \) equal to zero and solve for \( y \). This gives us \( y - 1 = 0 \) or \( y - 10 = 0 \), resulting in \( y = 1 \) and \( y = 10 \).

Step 4: Replace \( y \) with \( x^2 \) and Solve for \( x \)

Replace \( y \) with \( x^2 \) to solve for \( x \). This means solving the two equations \( x^2 = 1 \) and \( x^2 = 10 \).- For \( x^2 = 1 \), the solutions for \( x \) are \( x = \pm 1 \).- For \( x^2 = 10 \), the solutions for \( x \) are \( x = \pm \sqrt{10} \).

Step 5: Conclude with Complex Solutions

The solutions \( \pm 1 \) and \( \pm \sqrt{10} \) are the real solutions of the original equation. Because each is a real number, these are also the complex solutions that lie on the real axis of the complex plane.

Key Concepts

Quartic Polynomial

A quartic polynomial is a type of polynomial equation where the highest degree of the variable is four.
This means the equation takes the form \( ax^4 + bx^3 + cx^2 + dx + e = 0 \) where \( a eq 0 \).
Quartic polynomials can be more challenging to solve due to their higher degree compared to quadratic or cubic equations.
One might need to use different techniques such as factoring, substitution, or numerical methods in some cases.
In the exercise provided, the quartic polynomial is \( x^4 - 11x^2 + 10 = 0 \).
Notice that the equation has no cubic or linear terms, simplifying the task of solving it substantially.

Quadratic Substitution

Quadratic substitution is a method used to simplify solving higher-degree polynomials by turning them into more manageable quadratic equations.
This is particularly useful for polynomial equations that show a clear pattern or can be restructured into a quadratic form.
For the equation \( x^4 - 11x^2 + 10 = 0 \), a substitution such as \( y = x^2 \) is applied.
This substitution leads to a simpler equation \( y^2 - 11y + 10 = 0 \).

• Recognize patterns where a substitution will ease solving the equation.
• Apply the substitution uniformly across the equation.
• Transform the equation back using the original variable once the substituted variable is solved.

This transition into a quadratic form facilitates easier factoring and solution finding.

Factoring Quadratic Equations

Factoring is a method used to solve quadratic equations by expressing them as a product of simpler binomial expressions.
Factoring provides simpler solutions, particularly when roots are rational or integers.
For \( y^2 - 11y + 10 = 0 \), this equation was factored by identifying two numbers that multiply to 10 and add to -11, which are -1 and -10.
The factored form is \((y - 1)(y - 10) = 0 \).
Once factored, set each component \( y - 1 = 0 \) or \( y - 10 = 0 \) to solve for \( y \).
This gives roots \( y = 1 \) and \( y = 10 \), which relate back to our original variable after substitution reversal.

Complex Plane

The complex plane is a way of representing complex numbers graphically.
It involves visualizing real numbers as lying on a horizontal axis (real axis) and imaginary numbers on a vertical axis (imaginary axis).
This visualization allows easy comprehension of complex numbers, encompassing both real and imaginary components.
In this context, the solutions \( \pm 1 \) and \( \pm \sqrt{10} \) are entirely real, hence they lie on the real axis of the complex plane.
The complex plane helps to understand not only real solutions but also potential complex solutions where imaginary components are non-zero.
Even if solutions are real, placing them in a complex plane frames them within the broader and more comprehensive view of complex numbers.