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Chapter 1: Problem 76

Find the real solutions of each equation by factoring. $$ x^{4}-x^{2}=0 $$

Short Answer

Expert verified
The solutions are \(x = 0\), \(x = 1\), and \(x = -1\).

Step by step solution

01

- Factor out the common term

Notice that both terms on the left-hand side have a common factor of \(x^2\). Factor \(x^2\) out of the equation: \[ x^2(x^2 - 1) = 0 \]
02

- Set each factor to zero

Set each factor equal to zero: \[ x^2 = 0 \] and \[ x^2 - 1 = 0 \]
03

- Solve for \(x\)

Solve each equation separately. For \(x^2 = 0\): \[ x = 0 \]For \(x^2 - 1 = 0\): Add 1 to both sides: \[ x^2 = 1 \]Take the square root of both sides: \[ x = \frac{1}{2} \text{ or } x = -\frac{1}{2} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Real Solutions
Understanding 'real solutions' is important in algebra. Real solutions are the values of the variable that satisfy the equation, making the equation true when substituted back. In simpler terms, these are actual numbers you can plot on a number line that solve the equation. For instance, solving the equation in the example, finding real solutions means determining the exact values of 'x' that make the original equation true.
Factoring
Factoring is a method used to simplify polynomials, making them easier to solve. In the given exercise, both terms in the equation share a common factor, which is identified and factored out. Here the equation is: \[ x^4 - x^2 = 0 \]
Notice that both terms on the left-hand side have \(x^2\) in common. By factoring \(x^2\) out, we simplify the equation: \[ x^2(x^2 - 1) = 0 \] This breaks down the complex polynomial into simpler components that can be solved individually.
Solving Quadratic Equations
Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\). In our exercise, after factoring, we are left with simpler quadratic equations that can be solved separately. Setting each factor to zero:
  • \(x^2 = 0\)
  • \(x^2 - 1 = 0\)

Next, solve each equation independently. For \(x^2 = 0\), the solution is \(x = 0\). For \(x^2 - 1 = 0\), add 1 to both sides to get \(x^2 = 1\), then take the square root of both sides to find \(x = 1\) or \(x = -1\).
Algebra
Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. It is the foundation for solving equations such as the one given in the exercise. Fundamental operations include addition, subtraction, multiplication, and division, as well as advanced techniques like factoring and solving quadratic equations. Working with equations like \(x^4 - x^2 = 0\) involves understanding how to manipulate and transform expressions to isolate and solve for unknown variables.

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