Problem 37 - Algebra A Combined Function

Question

Solve each equation. $$ 4 x^{3}-x=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 0\), \(x = \frac{1}{2}\), and \(x = -\frac{1}{2}\).

Step 1: Factor the Equation

First, factor out the common term in the equation. Notice that both terms in the equation share a factor of \(x\). Factor \(x\) out of the equation: \(x(4x^2 - 1) = 0\).

Step 2: Apply the Zero Product Property

According to the zero product property, if a product of two factors equals zero, then at least one of the factors must be zero. Thus, set each factor equal to zero: \(x = 0\) or \(4x^2 - 1 = 0\).

Step 3: Solve the First Equation

The first factor is \(x = 0\). This equation is already solved, giving us the solution \(x = 0\).

Step 4: Solve the Second Equation

The second factor needs more work: \(4x^2 - 1 = 0\). Solve this equation by first adding 1 to both sides to get \(4x^2 = 1\). Then, divide both sides by 4 to isolate \(x^2\), yielding \(x^2 = \frac{1}{4}\).

Step 5: Take the Square Root

To solve for \(x\), take the square root of both sides of the equation \(x^2 = \frac{1}{4}\). Remember the square root will yield both positive and negative solutions: \(x = \pm\frac{1}{2}\).

Key Concepts

Factoring Equations

Factoring equations is an essential skill in algebra that helps simplify complex mathematical expressions. In the given exercise, we have the cubic equation, \[ 4x^{3} - x = 0. \]Our aim is to break down this expression into simpler parts so we can solve it easily.
One fundamental technique is identifying common factors in each term. Here, the common factor is \(x\). By factoring out \(x\), the equation transforms into:

• \( x(4x^2 - 1) = 0 \)

This process of factoring helps simplify the problem because it reduces a higher degree polynomial into a more manageable form by expressing it as a product of terms. Once an equation is factored, it's easier to apply further steps, like the Zero Product Property. Factoring is a basic algebraic operation but a very powerful one in solving equations.

Zero Product Property

The Zero Product Property is a foundational principle in algebra. It states that if a product of two terms equals zero, then at least one of the terms must be zero.
In our factored equation,

• \( x(4x^2 - 1) = 0 \)

the Zero Product Property allows us to split this into two simple equations:

• \( x = 0 \)
• \( 4x^2 - 1 = 0 \)

This property works because zero multiplied by any number is always zero. Therefore, solving each term separately for zero will help us find all possible solutions to the equation. It's a straightforward yet effective technique to isolate solutions that could be otherwise complicated to find.

Square Roots

Taking the square root is a common technique in solving quadratic equations and other algebraic expressions. In our example, one of the factoring results we got was:

• \( 4x^2 - 1 = 0 \)

We first isolate \(x^2\) by rearranging the equation:

• Add 1 to both sides: \(4x^2 = 1 \)
• Divide by 4: \(x^2 = \frac{1}{4} \)

To find \(x\), we take the square root of both sides:

• \( x = \pm \frac{1}{2} \)

Remember that when you take the square root of a number, you should consider both positive and negative roots. This is because both \( (\frac{1}{2})^2 \) and \( (-\frac{1}{2})^2 \) yield \(\frac{1}{4}\). By considering both solutions, you ensure that you capture all potential answers to the equation. This step effectively completes the solving process by providing the numerical solutions needed.