Problem 3 - A Graphical Approach to Precalculus with Limits

Question

Find all real solutions. Do not use a calculator. $$x^{4}-x^{2}=2 x^{2}+4$$

Step-by-Step Solution

Answer

Real solutions are \( x = 2 \) and \( x = -2 \).

Step 1: Rearrange the Equation

Start by bringing all terms to one side of the equation to set it to zero: \[ x^4 - x^2 - 2x^2 - 4 = 0 \] which simplifies to: \[ x^4 - 3x^2 - 4 = 0 \]

Step 2: Substitute a Variable for Simplification

To make this equation easier to solve, let \( y = x^2 \). Therefore, \[ x^4 = (x^2)^2 = y^2 \]. Substitute to get: \[ y^2 - 3y - 4 = 0 \]

Step 3: Solve the Quadratic Equation

This is now a quadratic equation, which can be factored as follows: \[ (y-4)(y+1) = 0 \]. Setting each factor equal to zero gives us: \[ y - 4 = 0 \] or \[ y + 1 = 0 \]. Thus, \( y = 4 \) or \( y = -1 \).

Step 4: Substitute Back to Solve for x

Recall \( y = x^2 \). Substitute back to find \( x \): For \( y = 4 \): \[ x^2 = 4 \] which implies \( x = 2 \) or \( x = -2 \). For \( y = -1 \): \( x^2 = -1 \), which is not possible for real numbers as the square of a real number cannot be negative. Hence, no solutions from this case.

Step 5: List the Real Solutions

The only real solutions from the previous steps are \( x = 2 \) and \( x = -2 \).

Key Concepts

Quadratic Equation

A quadratic equation is one that can be expressed in the form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, and \( x \) represents the unknown variable. These types of equations are called quadratic because the highest exponent of the variable \( x \) is 2.
Quadratic equations can appear in many ways, and solving them can often involve several methods such as factoring, using the quadratic formula, or completing the square. For the exercise at hand, after using variable substitution, we arrived at a more straightforward quadratic equation: \( y^2 - 3y - 4 = 0 \).

• This reduced equation can be approached using factoring, which is one of the simplest and most effective methods when applicable.
• The factoring method aims to express the equation as a product of two binomials, as seen with \( (y - 4)(y + 1) = 0 \).
• Once in this format, each factor can be set to zero to find the solutions for \( y \).

Mastering quadratic equations is fundamental as they form a basis for more complex algebraic concepts.

Variable Substitution

Variable substitution is a strategy used to simplify complex equations, essentially transforming them into a form that is easier to manage. In the given problem, the original equation \( x^4 - 3x^2 - 4 = 0 \) contains higher powers of \( x \), making it difficult to solve directly.
To simplify, we use substitution by letting \( y = x^2 \). This effectively turns our equation from a quartic equation (degree 4) into the quadratic \( y^2 - 3y - 4 = 0 \). This step drastically simplifies the problem.

• By substituting \( y = x^2 \), we broke down the quartic equation into a quadratic one, which is much more manageable.
• Once solved, we must remember to substitute back to the original variable to find the values of \( x \).
• Substitution is an excellent tool in problem-solving, particularly when faced with equations of higher degrees.

Through this substitution, we gain clearer insights and shift our focus from the original complex structure to an easier problem-solving pathway.

Real Solutions

Real solutions refer to solutions of an equation that can be represented as real numbers. Not all algebraic equations have real solutions, especially when involving square roots of negative numbers, which result in imaginary numbers instead.
In the exercise, after substituting back to the variable \( x \), we derived \( x^2 = 4 \) and \( x^2 = -1 \).

• For \( x^2 = 4 \), real solutions are found as \( x = 2 \) and \( x = -2 \), since these values satisfy the equation in real number terms.
• However, for \( x^2 = -1 \), there are no real solutions because a real number squared cannot yield a negative value.
• Discernment between real and non-real solutions is crucial, ensuring that solutions align with the problem's requirements.

Identifying real solutions plays a significant role in understanding the scope of any mathematical problem. Practicing this discrimination aids in effectively handling equations within the real number system.