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Magazine Chapter 7: Problem 38
Solve the given quadratic equations by factoring. $$x^{2}\left(a^{2}+2 a b+b^{2}\right)=x(a+b)$$
Short Answer
Expert verified
The solutions are \(x = 0\) and \(x = \frac{a+b}{a^{2}+2ab+b^{2}}\).
Step by step solution
01
Rearrange the Equation
To solve the quadratic equation, we start by rewriting it in a standard form. Given: \[ x^{2}(a^{2}+2ab+b^{2}) = x(a+b) \]Move all terms to one side of the equation:\[ x^{2}(a^{2}+2ab+b^{2}) - x(a+b) = 0 \]
02
Factor out the Common Factor
We notice that each term on the left side of the equation has a factor of \(x\). Factor \(x\) out of the equation:\[ x(x(a^{2}+2ab+b^{2}) - (a+b)) = 0 \]This simplifies to:\[ x((a^{2}+2ab+b^{2})x - (a+b)) = 0 \]
03
Solve Each Factor Separately
Set each factor in the equation to zero:1. \[ x = 0 \]2. \[ ((a^{2}+2ab+b^{2})x - (a+b)) = 0 \]
04
Solve the Second Factor
From the second equation:\[ ((a^{2}+2ab+b^{2})x - (a+b)) = 0 \] Rearrange to solve for \(x\):\[ (a^{2}+2ab+b^{2})x = a+b \]Divide both sides by \((a^{2}+2ab+b^{2})\) to get:\[ x = \frac{a+b}{a^{2}+2ab+b^{2}} \]
05
Combine Solutions
The solutions to the quadratic equation are obtained from both factors:\[ x = 0 \]\[ x = \frac{a+b}{a^{2}+2ab+b^{2}} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring Quadratics
Factoring quadratics is a method of expressing a quadratic equation as a product of its factors. This process is especially useful because it simplifies the solution of equations.
- A quadratic equation typically takes the form \( ax^2 + bx + c = 0 \) and is factored into two binomial expressions, if possible.
- The aim is to find two binomials that multiply to give the original quadratic expression.
- For the given problem, after rearranging and factoring out the common factor, we look to isolate \( x \) as a factor.
Standard Form of a Quadratic Equation
Understanding the standard form of a quadratic equation is crucial in solving these equations effectively. The standard form is \( ax^2 + bx + c = 0 \), where:
\[ x^2(a^2+2ab+b^2) - x(a+b) = 0 \]
Moving all terms to one side is the key step, as it displays the equation more clearly and sets the stage for effective factoring. This technique transforms it into a form that we are accustomed to dealing with.
- \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \).
- The term \( ax^2 \) is the quadratic term, \( bx \) is the linear term, and \( c \) is the constant term.
\[ x^2(a^2+2ab+b^2) - x(a+b) = 0 \]
Moving all terms to one side is the key step, as it displays the equation more clearly and sets the stage for effective factoring. This technique transforms it into a form that we are accustomed to dealing with.
Solving Quadratic Equations
Once a quadratic equation is arranged, solving it involves finding the values of the variable that make the equation true. Here鈥檚 how we solve it when factored:
- We begin by setting each factor to zero independently because of the zero-product property, which states that if \( ab = 0 \), then \( a = 0 \) or \( b = 0 \).
- In this case, setting up \( x = 0 \) offers our first solution directly.
- For the second term, \( ((a^2+2ab+b^2)x - (a+b)) = 0 \), we further simplify to isolate \( x \).
- Solving for \( x \) involves rearranging to find \( x = \frac{a+b}{a^2+2ab+b^2} \), which provides another point of solution.
Algebraic Manipulation
Algebraic manipulation is a crucial skill in rearranging and solving quadratic equations. This involves using mathematical operations strategically to simplify expressions and solve equations.
- In our case, we initially rearranged the equation to bring all terms to one side. This initial algebraic rearrangement puts the equation into a solvable format.
- Further manipulation includes factoring out common factors, like pulling out \( x \), to simplify the equation further.
- Finally, dividing each side of our simplified equation can isolate the variable, solving the quadratic completely.
