Chapter 10: Problem 45
Solve the equation. Tell which solution method you used. \(10 x^{3}-290 x^{2}+620 x=0\)
Short Answer
Expert verified
The solutions to the equation \(10 x^{3} - 290 x^{2} + 620 x = 0\) are \(x = 0\), \(x = 23\), and \(x = 27\). The equation is solved using a factorization and quadratic formula method.
Step by step solution
01
Identify the coefficients
In the given equation \(10 x^{3} - 290 x^{2} + 620 x = 0\), the coefficients of \(x\) are \(a = 10\), \(b = -290\), and \(c = 620\).
02
Factor out the Greatest Common Factor (GCF)
The coefficients have a common factor of \(10x\), so factor this out to simplify the equation into \(10x (x^{2} - 29x + 62) = 0\).
03
Use the Zero Product Property
The zero product property states that if a product of factors equals zero, then at least one of the factors must be zero. Setting \(10x = 0\) gives \(x = 0\). Setting \(x^{2} - 29x + 62 = 0\) will require solving a quadratic equation.
04
Solve the Quadratic Equation
To solve the quadratic equation \(x^{2} - 29x + 62 = 0\), use the quadratic formula \(x = [-b ± sqrt(b^{2} - 4ac)] / 2a\). Substituting \(a = 1\), \(b = -29\), and \(c = 62\) gives \(x = [29 ± sqrt((-29)^{2} - 4*1*62)] / 2*1\), which simplifies to \(x = 23\) and \(x = 27\).
05
Write the Solutions
The solutions to the equation \(10 x^{3} - 290 x^{2} + 620 x = 0\) are therefore \(x = 0\), \(x = 23\), and \(x = 27\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring Cubic Polynomials
When faced with a cubic polynomial, such as in the exercise, one of the first strategies is to identify and factor out the greatest common factor, if there is any. This simplifies the expression and may reduce the cubic equation to a quadratic one.
After factoring out the common factor, the next step usually involves looking for ways to factor the remaining polynomial. This could involve methods such as grouping, using the sum or difference of cubes, or employing synthetic division if a rational root can be presumed. In some cases, one might have to resort to numerical methods or formulas specific for cubic equations if the polynomial cannot be factored over the rational numbers easily.
In our example, the process begins by simplifying the cubic polynomial by factoring, thus reducing the complexity of the problem and bringing us a step closer to finding the solution.
After factoring out the common factor, the next step usually involves looking for ways to factor the remaining polynomial. This could involve methods such as grouping, using the sum or difference of cubes, or employing synthetic division if a rational root can be presumed. In some cases, one might have to resort to numerical methods or formulas specific for cubic equations if the polynomial cannot be factored over the rational numbers easily.
In our example, the process begins by simplifying the cubic polynomial by factoring, thus reducing the complexity of the problem and bringing us a step closer to finding the solution.
Zero Product Property
A fundamental tool in algebra is the zero product property, which tells us that if the product of several factors equals zero, then at least one of those factors must be zero on its own. This property is invaluable when solving equations because it allows us to isolate each factor to find possible solutions.
In the context of the given exercise, once the greatest common factor was factored out, we applied the zero product property. By setting each factor equal to zero, we uncovered the roots of the equation. This property simplifies the solving process as we can handle each factor one at a time, and it is particularly useful when dealing with equations that can be factored into linear terms.
In the context of the given exercise, once the greatest common factor was factored out, we applied the zero product property. By setting each factor equal to zero, we uncovered the roots of the equation. This property simplifies the solving process as we can handle each factor one at a time, and it is particularly useful when dealing with equations that can be factored into linear terms.
Quadratic Formula
For quadratic equations, one that cannot be readily factored, the quadratic formula is a reliable tool. It is given by the formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\), where \(a\), \(b\), and \(c\) are the coefficients of the terms \(ax^2\), \(bx\), and \(c\), respectively.
The beauty of the quadratic formula lies in its universality—it works for any quadratic equation. It allows us to find the roots of the equation by substituting the coefficients into the formula. As we saw in our exercise, after factoring out the GCF and applying the zero product property, the quadratic formula provided the remaining solutions to the equation. It's an essential tool for students to master as it provides a methodical way of identifying solutions to quadratic equations.
The beauty of the quadratic formula lies in its universality—it works for any quadratic equation. It allows us to find the roots of the equation by substituting the coefficients into the formula. As we saw in our exercise, after factoring out the GCF and applying the zero product property, the quadratic formula provided the remaining solutions to the equation. It's an essential tool for students to master as it provides a methodical way of identifying solutions to quadratic equations.
Greatest Common Factor
Identifying the greatest common factor (GCF) among the terms of a polynomial is a vital initial step in simplifying complex problems. It's the largest polynomial that divides each of the terms without leaving a remainder, which can make the equation easier to solve.
In our example, we extracted the GCF of 10x from each term, which simplified the cubic polynomial to a quadratic one, making it more manageable. Factoring out the GCF not only can make subsequent steps of solving the polynomial more straightforward even in more complex scenarios but also teaches students about the importance of reducing expressions to their simplest form before proceeding with further problem-solving strategies.
In our example, we extracted the GCF of 10x from each term, which simplified the cubic polynomial to a quadratic one, making it more manageable. Factoring out the GCF not only can make subsequent steps of solving the polynomial more straightforward even in more complex scenarios but also teaches students about the importance of reducing expressions to their simplest form before proceeding with further problem-solving strategies.