Problem 64 - Algebra 1: Concepts and Skills

Question

Solve the equation by factoring. $$ x^{2}-25=0 $$

Step-by-Step Solution

Answer

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

Step 1: Transform the Equation

First, transform the equation to make it looks like a difference of squares equation: \(x^{2}-25=0\).

Step 2: Apply the Difference of Squares Rule

Now, apply the difference of squares rule, which says that \(a^{2} - b^{2} = (a - b)(a + b)\). In this case, \( a = x \) and \( b = 5 \). Thus, the equation \(x^{2}-25 = 0\) can be written as \((x - 5)(x + 5) = 0\).

Step 3: Find the roots

Next, set each factor equal to zero and solve for x to find the roots of the equation: \(x - 5 = 0\) and \(x +5 = 0\). After solving these two equations, get \(x=5\) and \(x=-5\).

Key Concepts

Difference of Squares

The difference of squares is a convenient mathematical identity that helps simplify certain expressions. When you see two squared terms separated by a subtraction sign, you should immediately think of this identity, which states:

• \(a^2 - b^2 = (a - b)(a + b)\)
• Here, \(a\) and \(b\) are any expressions or numbers.

This helps turn a difference of squares expression into a product of two binomials, which is often easier to manipulate or solve.
In the example equation \(x^2 - 25\), we recognize it as a difference of squares because the problem can be expressed as \(x^2 - 5^2\). Using the identity, we rewrite it as:

• \((x - 5)(x + 5)\)
• This transformation is the key step in preparing to solve the equation by factoring.

Roots of Quadratic Equations

To solve quadratic equations like \(x^2 - 25 = 0\), you first find the roots, also known as solutions. The roots of the equation are the values of \(x\) for which the equation is satisfied.
Once you have factored the equation (e.g., \((x - 5)(x + 5) = 0\)), the next step is relatively simple. Since this is a zero-product situation, any value that makes either factor zero will solve the equation.
Set each factor equal to zero:

• \(x - 5 = 0\)
• \(x + 5 = 0\)

Solving these gives you:

• \(x = 5\)
• \(x = -5\)

These are the roots, meaning if you plug these values back into the original equation \(x^2 - 25 = 0\), they should satisfy it, confirming our solutions.

Solving Equations by Factoring

Factoring is a powerful method for solving quadratic equations. It involves expressing a quadratic expression as a product of binomials. By identifying patterns such as the difference of squares, you can rewrite the equation in a more solvable form.
For our specific equation \(x^2 - 25 = 0\), factoring it gives us \((x - 5)(x + 5) = 0\). This method leverages the "zero product property," which states that if a product of factors equals zero, at least one of the factors must be zero.

• Factor the equation into two binomials.
• Set each factor equal to zero.
• Solve each resulting simple equation.

Factoring is especially useful when equations fit recognizable patterns like the difference of squares. This makes it efficient and effective to find solutions without needing to graph or guess. So remember, when faced with a straightforward quadratic, check if it can be factored for a quick solution.