Problem 17 - A Graphical Approach to College Algebra

Question

Find all complex solutions of each equation. $$x^{4}-11 x^{2}+10=0$$

Step-by-Step Solution

Answer

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

Step 1: Recognize the Structure

Notice that the given equation is a polynomial of degree 4: \(x^4 - 11x^2 + 10 = 0\). This equation can be approached by using substitution to make it easier to solve.

Step 2: Use a Substitution

Let \(y = x^2\). This substitution simplifies the polynomial to \(y^2 - 11y + 10 = 0\), which is a quadratic equation.

Step 3: Solve the Quadratic Equation

Use the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve for \(y\). Here \(a = 1\), \(b = -11\), and \(c = 10\). Plugging these values in, we get:\[y = \frac{11 \pm \sqrt{121 - 40}}{2} = \frac{11 \pm \sqrt{81}}{2} = \frac{11 \pm 9}{2}\]This gives us \(y = 10\) or \(y = 1\).

Step 4: Substitute Back to Find \(x\)

Recall that \(y = x^2\). Substitute back the \(y\) values to find \(x\):1. If \(y = 10\), then \(x^2 = 10\), thus \(x = \sqrt{10}\) or \(x = -\sqrt{10}\).2. If \(y = 1\), then \(x^2 = 1\), thus \(x = 1\) or \(x = -1\).

Step 5: List All Solutions

Combine all the solutions derived from the substitution back step. The complex solutions of the original equation are \(x = \sqrt{10}, -\sqrt{10}, 1, -1\).

Key Concepts

Polynomial Equations

A polynomial equation is an algebraic equation of the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0\), where the exponents \(n, n-1, ... , 0\) are non-negative integers, and \(a_n, a_{n-1}, ..., a_0\) are coefficients. Polynomials are characterized by their degree, which is the highest exponent of the variable \(x\).
For example, in the equation given \(x^4 - 11x^2 + 10 = 0\), the highest exponent is 4, making it a fourth-degree polynomial.
This type of equation may have up to four solutions, which can be real or complex numbers.

Polynomials can often be challenging to solve directly, particularly those of higher degrees. When confronted with such challenges, mathematicians employ various strategies like factoring, using the substitution method, or applying the quadratic formula to simplify the problem into a more manageable form. Understanding these approaches is crucial for successfully solving polynomial equations.

Quadratic Formula

The quadratic formula is a powerful tool used for finding solutions to quadratic equations of the form \(ax^2 + bx + c = 0\).
The formula is expressed as:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
It provides the solutions for \(x\), factoring the discriminant \(b^2 - 4ac\).
This discriminant determines the nature of the roots of the polynomial:

• If \(b^2 - 4ac > 0\), there are two distinct real solutions.
• If \(b^2 - 4ac = 0\), there are two identical real solutions (a repeated root).
• If \(b^2 - 4ac < 0\), there are two complex solutions.

In the problem above, leveraging the quadratic formula, the equation \(y^2 - 11y + 10 = 0\) was solved.
Plugging the values \(a = 1\), \(b = -11\), \(c = 10\), into the quadratic formula provided both possible values for \(y\), leading ultimately to finding all solutions for \(x\).
This method not only simplifies solving quadratics but also extends to solve certain higher-degree polynomial equations when substitution is applied.

Substitution Method

The substitution method is a strategic approach in algebra that simplifies complex equations by transforming them into simpler ones.
It involves replacing a part of the equation with a new variable to reduce its complexity.

In the equation \(x^4 - 11x^2 + 10 = 0\), substituting \(y = x^2\) transformed this quartic (degree 4) equation into a quadratic equation \(y^2 - 11y + 10 = 0\).
This is easier to manage and solve with the quadratic formula. Once the solutions for \(y\) are found, these are then translated back into the terms of the original variable \(x\).

The substitution method is useful because:

• It can simplify high-degree equations to lower-degree ones.
• It helps eliminate complicated expressions, making manipulation easier.
• It can lead directly to applying other straightforward methods, like the quadratic formula.

Recognizing when to use substitution can significantly streamline solving polynomial equations and is a crucial skill in algebra.