Problem 41 - Introductory Algebra for College Students

Question

Use factoring to solve each quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\) -intercepts. $$81 x^{2}=25$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 5/9\) and \(x = -5/9\)

Step 1: Rearrange the Equation

First, rearrange the quadratic equation to the Standard Form \(ax^2 + bx + c = 0\). In this case, the given equation would be \(81x^2 - 25 = 0\)

Step 2: Factoring using difference of squares

Rewrite the equation as a difference of squares. The equation could be rewritten as \((9x)^2 - 5^2 = 0\)

Step 3: Apply the difference of squares formula

Apply the Formula \(a^2 - b^2 = (a-b)(a+b)\), yielding \( (9x-5)(9x+5) = 0 \)

Step 4: Find x

Set each factor equal to zero and solve for x. This results in \(9x - 5 = 0\) or \(9x + 5 = 0\), which gives \(x = 5/9\) for the first equation and \(x = -5/9\) for the second equation

Key Concepts

Difference of Squares

The difference of squares is a special factoring technique used in algebra to simplify expressions that take the form \(a^2 - b^2\). This can be factored into \((a - b)(a + b)\). It is called "difference of squares" because it involves two square numbers separated by a subtraction sign.

Let's break down the main parts of this tactic:

• Identifying Squares: Each term in the equation should be a perfect square. In our exercise, \(81x^2\) is \((9x)^2\) and 25 is \(5^2\).
• Formula Application: Once identified, apply the formula: \(a^2 - b^2 = (a - b)(a + b)\). For \(81x^2 - 25 = 0\), it simplifies to \((9x - 5)(9x + 5) = 0\).

This approach is very useful because it reduces the problem to solving two linear equations. Recognizing when an expression is a difference of squares is crucial for solving equations quickly and efficiently.

Quadratic Formula

The quadratic formula is a powerful tool in algebra for solving any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is given by:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]This means you can find the solutions (or roots) of the equation as long as you know the coefficients \(a\), \(b\), and \(c\).

Here is how it works:

• Plug in Values: Insert the values of \(a\), \(b\), and \(c\) from your equation into the formula.
• Solve for Roots: Simplify the expression to find the x-values (roots) of the equation.

In our given problem, the equation \(81x^2 - 25 = 0\) can be translated into \(a = 81, b = 0, c = -25\). While not necessary for this exercise, the quadratic formula provides another method to verify the solutions derived from factoring.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of \(x\) that make the equation true. There are several methods available to solve them effectively.

The common methods include:

• Factoring: As seen with the difference of squares, factoring is a straightforward method when applicable. It was used successfully in the exercise to solve \(81x^2 - 25 = 0\).
• Quadratic Formula: This is a universal method that works on any quadratic equation. It's highly useful when other factoring methods are difficult.
• Graphing: By plotting the quadratic equation on a graph, you can find the \(x\)-intercepts, which correspond to the solution(s) of the equation.

Each method has its advantages, depending on the complexity of the equation. By mastering different methods, you gain flexibility and efficiency in solving quadratic equations, allowing for verification and deeper understanding of algebraic solutions.