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Chapter 1: Problem 77

Find the real solutions of each equation by factoring. $$ 4 x^{3}=3 x^{2} $$

Short Answer

Expert verified
The solutions are \(x = 0\) and \(x = \frac{3}{4}\).

Step by step solution

01

Move All Terms to One Side

To set up the equation for factoring, move all terms to one side by subtracting \(3x^{2}\) from both sides of the equation:\[4x^{3} - 3x^{2} = 0\]
02

Factor Out the Greatest Common Factor (GCF)

Identify and factor out the greatest common factor from the expression:\[x^{2}(4x - 3) = 0\]
03

Set Each Factor to Zero

To find the solutions, set each factor equal to zero and solve for \(x\):\[x^{2} = 0 \]This implies \(x = 0\). Next, solve the linear equation:\[4x - 3 = 0\]\[4x = 3\]\[x = \frac{3}{4}\]
04

Combine the Solutions

Combine both solutions obtained from setting each factor to zero:\[x = 0 \text{ and } x = \frac{3}{4}\]

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Key Concepts

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

solving cubic equations
Cubic equations are polynomial equations where the highest degree term is raised to the power of three. They can be challenging to solve, but breaking them down into smaller steps can simplify the process. Let's take the equation \(4x^{3} = 3x^{2}\). To solve this:

  • First, move all terms to one side of the equation. This converts it to: \(4x^{3} - 3x^{2} = 0\)
  • Next, factor out the greatest common factor (GCF) from the terms.
This transformed equation helps us set up the problem for factoring. By factoring out common factors, we reduce the complexity and make it easier to find the roots or solutions.
greatest common factor
The greatest common factor (GCF) is the largest term that divides two or more numbers without leaving a remainder. In polynomial equations, identifying the GCF simplifies factoring. For our equation: \(4x^{3} - 3x^{2} = 0\):

  • Look at each term: \(4x^{3}\) and \(3x^{2}\).
  • The common factor between them is \(x^{2}\).
Now factor out \(x^{2}\) from the equation: \(x^{2}(4x - 3) = 0\). This step reduces the problem to simpler terms, allowing us to solve for \(x\). Each factor gives us a potential solution. Set \(x^{2} = 0\) and \(4x - 3 = 0\) to find all possible solutions for \(x\).
real solutions
Real solutions are the values of \(x\) that satisfy the equation within real numbers. For our equation \(4x^{3} = 3x^{2}\), after factoring, we have:\(x^{2}(4x - 3) = 0\). Set each factor equal to zero and solve:

  • \(x^{2} = 0\) which gives \(x = 0\)
  • \(4x - 3 = 0\), solve for \(x\): \(4x = 3\) so \(x = \frac{3}{4}\)
Combine these solutions: \(x = 0\) and \(x = \frac{3}{4}\). Both are real solutions for the original cubic equation. This process helps ensure that all potential solutions are considered and verified.

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Most popular questions from this chapter

Find the real solutions, if any, of each equation. Use any method. $$ 9 x^{2}-12 x+4=0 $$

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