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Magazine Chapter 7: Problem 68
The following equations are not quadratic but can be solved by factoring and applying the zero product rule. Solve each equation. $$14 a^{2}-49 a=a^{3}$$
Short Answer
Expert verified
The short answer is that the possible values for 'a' that satisfy the given equation are \(a = 0\) and \(a = 7\).
Step by step solution
01
Rearrange the equation
To begin the process of factoring and applying the zero product rule, let's first rearrange the equation so that all terms are on one side, giving us a polynomial that is equal to zero. From the given equation:
\(14a^2 - 49a = a^3\)
Subtract \(a^3\) from both sides to obtain:
\(-a^3 + 14a^2 - 49a = 0\)
02
Factor out the common factor
Now we need to factor out the common factor, which in this case is 'a'. Factoring 'a' out of the rearranged equation gives us:
\(a (-a^2 + 14a - 49) = 0\)
03
Factoring the quadratic expression
The next step is to focus on factoring the remaining part of the expression, which is a quadratic expression: \(-a^2 + 14a - 49\). In order to make factoring easier, let's first factor out the negative sign, resulting in:
\(a (a^2 - 14a + 49) = 0\)
Now we have to find two numbers that when multiplied equal (1)(49), and when added equal -14. These numbers are -7 and -7. Factoring the quadratic expression results in:
\(a (a - 7)^2 = 0\)
04
Applying the Zero Product Rule
Now that we have factored the expression, we can apply the zero product rule. According to this rule, if the product of two factors is zero, then at least one of them must be equal to zero. In this case, we have two factors: 'a' and \((a - 7)^2\).
Set each factor equal to zero and solve for 'a':
1. \(a = 0\)
2. \((a - 7)^2 = 0\)
For the second equation, take the square root of both sides:
\(a - 7 = \pm \sqrt{0}\)
Since the square root of 0 is 0, the equation simplifies to:
\(a - 7 = 0\)
So, the only other solution is:
\(a = 7\)
05
State the solutions
We found two possible solutions for the variable 'a' using factoring and the zero product rule. These solutions are:
\(a = 0\)
\(a = 7\)
These are the two possible values for 'a' that satisfy the given equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
zero product rule
The zero product rule is a fundamental principle in algebra that enables us to find solutions for products equaling zero. Simply put, if the product of two or more terms is zero, at least one of the terms must also be zero. For instance, if you have a product like \(xy = 0\), this means either \(x = 0\) or \(y = 0\) (or both). This rule is particularly useful when dealing with equations that have been factored, as it allows you to break the problem down into smaller, more manageable pieces.
- First, ensure the equation is equal to zero and properly factored.
- Set each factor that results from the factoring step equal to zero.
- Solve each resulting simple equation to find potential solutions.
quadratic expression
A quadratic expression is any polynomial that can be written in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(aeq 0\). These expressions are central to understanding algebra and appear frequently in various problems across mathematics. In this particular exercise, we encountered the quadratic expression \(-a^2 + 14a - 49\).
When you handle quadratic expressions, the key is recognizing patterns or using techniques such as factoring, completing the square, or the quadratic formula. For factoring, one effective method is determining two numbers that multiply to give the product \(a\cdot c\) and add to give \(b\). In our example, these numbers were \(-7\) and \(-7\), allowing us to factor the expression into \((a - 7)^2\).
When you handle quadratic expressions, the key is recognizing patterns or using techniques such as factoring, completing the square, or the quadratic formula. For factoring, one effective method is determining two numbers that multiply to give the product \(a\cdot c\) and add to give \(b\). In our example, these numbers were \(-7\) and \(-7\), allowing us to factor the expression into \((a - 7)^2\).
polynomial rearrangement
Rearranging polynomials involves structuring the terms of a polynomial equation to facilitate the application of algebraic techniques like factoring. In the given problem, the original equation was not arranged as a polynomial equal to zero. This required rearrangement: \(14a^2 - 49a = a^3\) was transformed into \(-a^3 + 14a^2 - 49a = 0\).
- Move all terms to one side of the equation to create a standard polynomial form, \(Ax^{n} + Bx^{m} + \ldots = 0\).
- Simplify the equation by combining like terms if needed.
- This step is essential as it sets the stage for further operations like factoring.
solutions of equations
Finding solutions to equations is a critical skill in algebra, particularly for understanding how different values satisfy an equation. In our exercise, we found the solutions for \(a\) in the equation \(14a^2 - 49a = a^3\) through a series of steps.
After rearranging and factoring, we used the zero product rule to identify potential solutions: \(a = 0\) and \(a = 7\).
After rearranging and factoring, we used the zero product rule to identify potential solutions: \(a = 0\) and \(a = 7\).
- Verify your solutions by substituting them back into the original equation to ensure the equality holds.
- Understanding the context of the solutions can help鈥攕ometimes equations have multiple solutions, and sometimes only one.
