Problem 38 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations by factoring. $$x^{2}\left(a^{2}+2 a b+b^{2}\right)=x(a+b)$$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = \frac{a+b}{a^{2}+2ab+b^{2}}\).

Step 1: 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:\[ x^{2}(a^{2}+2ab+b^{2}) - x(a+b) = 0 \]

Step 2: 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:\[ x((a^{2}+2ab+b^{2})x - (a+b)) = 0 \]

Step 3: Solve Each Factor Separately

Set each factor in the equation to zero:1. \[ x = 0 \]2. \[ ((a^{2}+2ab+b^{2})x - (a+b)) = 0 \]

Step 4: 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:\[ x = \frac{a+b}{a^{2}+2ab+b^{2}} \]

Step 5: Combine Solutions

The solutions to the quadratic equation are obtained from both factors:\[ x = 0 \]\[ x = \frac{a+b}{a^{2}+2ab+b^{2}} \]

Key Concepts

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.

Once factored, each factor can be solved separately to find the values of \( x \) that satisfy the equation. This method relies heavily on identifying common terms, and sometimes it requires creative algebraic manipulation to notice intricate factorizations.

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:

• \( a \), \( b \), 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.

In the given problem, the equation is initially not in standard form. To fix this, the equation is rearranged so all terms are on one side:
\[ x^2(a^2+2ab+b^2) - x(a+b) = 0 \]
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’s 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.

Both solutions must satisfy the original equation, hence validating our algebraic manipulation.

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.

This process involves understanding and applying various algebraic rules and properties correctly, ensuring that errors are minimized. A methodical approach in transforming and simplifying equations is vital for successful problem-solving.