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Chapter 10: Problem 49

Solve the equation. Tell which solution method you used. \(24 x^{3}+18 x^{2}-168 x=0\)

Short Answer

Expert verified
The solutions for the cubic equation \(24 x^{3}+18 x^{2}-168 x=0\) are \(x = 0\), \(x = \frac{7}{4}\), \(x = -4\). The solution method used was factoring.

Step by step solution

01

Simplifying the equation

Notice that each term in the equation \(24 x^{3}+18 x^{2}-168 x=0\) can be divided by 6. This gives us the simplified equation: \(4x^{3} + 3x^{2} - 28x = 0\).
02

Factoring out common terms

Now, notice that each term in the simplified equation has \(x\) as a common factor. Factor out \(x\) from each term to give: \(x(4x^{2} + 3x - 28) = 0\).
03

Factorizing the quadratic equation

We now have to solve the quadratic equation \(4x^{2} + 3x - 28\) for \(x\). This equation can be factorized to \((4x -7)(x + 4) = 0\). Thus the equation becomes \(x(4x - 7)(x + 4) = 0\).
04

Finding the solutions

Equation \(x(4x - 7)(x + 4) = 0\) will equal zero when either of \(x\), \(4x - 7\) or \(x + 4\) equals zero. The solutions for the equation are therefore \(x = 0\), \(x = \frac{7}{4}\), \(x = -4\).

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Key Concepts

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

Factoring
Factoring is a crucial method for solving polynomial equations. It involves breaking down an expression into simpler terms (factors) that, when multiplied together, give the original expression. In our example, the equation starts as \(24x^3 + 18x^2 - 168x = 0\). By simplifying each term, you notice they can all be divided by 6. This reduces the equation to \(4x^3 + 3x^2 - 28x = 0\).

Next, you observe that each term contains an \(x\), allowing you to factor it out: \(x(4x^2 + 3x - 28) = 0\). Factoring completely might involve identifying patterns such as difference of squares or using methods like grouping.
  • Always look for common factors first, this simplifies the equation.
  • Check if the leftover polynomial is factorable using standard methods.
  • Remember that factoring transforms complex equations into simpler products.
Quadratic Equations
A quadratic equation is any equation that can be rearranged in the form \(ax^2 + bx + c = 0\). In our context, the reduced polynomial \(4x^2 + 3x - 28 = 0\) is a quadratic. Solving quadratics often involves factoring, completing the square, or using the quadratic formula.

In this exercise, after factoring out the common \(x\), we still have the quadratic \(4x^2 + 3x - 28\). To factor this, you look for two numbers that multiply to \(a \times c\) (in this case \(-112\)) and add to \(b\) (which is 3 here). Those numbers are 16 and -7. This guides you to rewrite and factor as \((4x - 7)(x + 4) = 0\).
  • Understand different methods to solve depending on the form it takes.
  • Factoring involves finding pairs of numbers that work with both product and sum conditions.
  • Ensure each step logically connects to the next to avoid errors.
Solution Methods
When dealing with polynomials, especially cubic equations like \(24x^3 + 18x^2 - 168x=0\), there are systematic solution methods useful. The process usually involves simplifying, factoring, and solving the resulting equations step-by-step.

In this problem, you first simplify by factoring out a common term, then deal with a quadratic, ultimately resulting in factors. To find solutions from \(x(4x - 7)(x + 4) = 0\), set each factor equal to zero: \(x = 0\), \(4x - 7 = 0\) resulting in \(x = \frac{7}{4}\), and \(x + 4 = 0\) giving \(x = -4\).
  • Choose a method that best suits the form of the equation.
  • Setting each factor to zero leverages the zero-product property.
  • Always verify the solutions back in the original equation.

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Most popular questions from this chapter

Use factoring to solve the equation. Use a graphing calculator to check your solution if you wish. $$50 x^{2}+60 x+18=0$$

Solve the equation. \(|x+6|=13\)

Factor the expression. Tell which special product factoring pattern you used. $$x^{2}+\frac{2}{3} x+\frac{1}{9}$$

Which of the following is the solution of the equation \(-4 x^{2}+24 x-36=0 ?\) A) -6 B) -3 C) 2 D) 3

Decide whether or not the ordered pair is a solution of the system of linear equations. $$\begin{aligned} &-5 x-8 y=28\\\ &9 x-2 y=48 \quad(4,-6) \end{aligned}$$
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