Problem 58 - Contemporary Precalculus

Question

Find all real solutions of the equation exactly. $$x^{4}-2 x^{2}+1=0$$

Step-by-Step Solution

Answer

Question: Find all real solutions for the equation \(x^4 - 2x^2 + 1 = 0\). Answer: The real solutions for the given equation are \(x = -1, 1\).

Step 1: Substitution

Let's set \(y = x^2\). Then our equation turns into a quadratic equation: $$y^2 - 2y + 1 = 0$$

Step 2: Solve the quadratic equation

Now, we have to solve the quadratic equation for y: $$y^2 - 2y + 1 = (y - 1)^2 = 0$$ The solution is \(y = 1\).

Step 3: Substitute back and solve for x

Since we have found the value of \(y\), we can substitute it back into the original substitution we made (\(y= x^2\)): $$x^2 = 1$$ So, the solutions for x are: $$x = \pm \sqrt{1} = \pm 1$$

Step 4: Present the final solutions

The real solutions for the given equation are: $$x = -1, 1$$

Key Concepts

Quadratic Substitution Method

When tackling complex equations, breaking them down into simpler components can be incredibly helpful. The quadratic substitution method embodies this approach. In essence, it transforms a higher-order polynomial into a quadratic equation, which most students find easier to manage.

Consider the given exercise where we encounter a fourth-degree polynomial equation: \[x^{4} - 2x^{2} + 1 = 0\]. Our first step is to notice a pattern that resembles a quadratic, and we use substitution to exploit this resemblance. By setting \(y = x^2\), the equation becomes \[y^2 - 2y + 1 = 0\], which is notably easier to solve.

This method not only makes the solving process more approachable but also allows for the application of familiar techniques, such as factoring, completing the square, or using the quadratic formula. Substitution is powerful because it can turn what seems like an impenetrable problem into a format with ready-to-use methods for finding solutions.

Finding Real Solutions

The quest to find real solutions to an equation is like a detective uncovering clues to solve a mystery. Real solutions are the x-values that make the equation true, and these can include positive numbers, negative numbers, and zero. The equation from the exercise \[x^4 - 2x^2 + 1 = 0\] presents a unique challenge because it's not immediately clear what the values for \(x\) might be.

After applying the quadratic substitution method, we get a simpler quadratic equation. By solving this new equation, we ascertain the value(s) of the substitution variable, which in this case is \(y\). Next, a crucial step involves translating these solutions back to the original variable, \(x\). This step can unveil more than one potential solution for \(x\) because when we reverse the substitution (taking the square root of a number), we must consider both the positive and negative possibilities. It is always vital to check all possible values to ensure they are indeed real solutions that satisfy the original equation.

Quadratic Formula

The quadratic formula is a reliable and universal method for solving quadratic equations. It is expressed as:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\].This elegantly conveys the solutions to any quadratic equation of the form \(ax^2 + bx + c = 0\). In our exercise, after substituting \(y = x^2\), we didn't need to use the quadratic formula because we could factor easily. However, not all quadratic equations are so cooperative.

When dealing with equations that are not readily factorable, or if finding factors quickly becomes complicated, the quadratic formula is an invaluable tool. It systematically provides the solutions by plugging the coefficients of the quadratic equation into the formula. The discriminant, \(b^2 - 4ac\), within the formula determines the nature of the solutions. When the discriminant is positive, we get two distinct real solutions; if it's zero, there's exactly one real solution; and a negative discriminant indicates no real solutions, only complex ones. The quadratic formula provides a surefire way to proceed when the path to the solution is not immediately clear.