Problem 23 - Algebra 1: Concepts and Skills

Question

Solve the equation or write no real solution. Write the solutions as integers if possible. Otherwise, write them as radical expressions. $$ x^{2}=225 $$

Step-by-Step Solution

Answer

The solutions are x = 15 and x = -15.

Step 1: Rewrite the equation

First, rewrite the given equation as \(x^{2}-225=0\) to see it more clearly as difference of squares.

Step 2: Apply the square root property

Secondly, apply the square root property to solve for x. This property tells us that if \(x^2 = c\), then \(x = \sqrt{c}\) or \(x = - \sqrt{c}\). Therefore the equation can be written as \(x = \sqrt{225}\) or \(x = - \sqrt{225}\).

Step 3: Simplify and solve for x

Lastly, simplify and solve for x. Since square root of 225 equals 15, the two solution for the equation are \(x = 15\) and \(x = -15\).

Key Concepts

Difference of Squares

The difference of squares is a specific way of factoring where you have two squares subtracted from one another. It follows the pattern \( a^2 - b^2 = (a + b)(a - b) \). This concept is essential when dealing with quadratic equations like \( x^2 = 225 \), which can be rewritten as \( x^2 - 225 = 0 \). This form reveals itself as a difference of squares because it can be seen as \( x^2 - 15^2 = 0 \). Here:

• \( a = x \)
• \( b = 15 \)

Recognizing it as a difference of squares allows us to factor the equation into \( (x + 15)(x - 15) = 0 \). This step helps in setting each part of the factored equation equal to zero to find potential solutions. Each factor leads us closer to determining the values of \( x \) that satisfy the equation.

Square Root Property

The square root property simplifies solving equations that have the form \( x^2 = c \). According to this property, you can solve the equation by taking the square root of both sides. This results in two possible solutions: one positive and one negative. For example, if we have \( x^2 = 225 \), applying the square root property means calculating \( x = \pm \sqrt{225} \). Here:

• \( x = \sqrt{225} \) which simplifies to \( x = 15 \)
• \( x = -\sqrt{225} \) which simplifies to \( x = -15 \)

It's crucial because the square root property provides both the positive and negative solutions to quadratic equations, reflecting that both values of \( x \) satisfy \( x^2 = c \). This step ensures that we explore all possible solutions and not just the positive one.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. In our case, we focused on square roots. When a number is not a perfect square, its square root is represented as a radical expression, like \( \sqrt{c} \). However, for our solution \( \sqrt{225} \), since 225 is a perfect square (being equal to \( 15^2 \)), the radical form simplifies neatly to an integer, 15.
Radical expressions maintain their importance especially when solutions do not neatly reduce to integers. If \( x^2 = c \) had \( c \) as a non-perfect square, then \( x = \sqrt{c} \) would remain a radical.Understanding radical expressions can help:

• Represent solutions to equations where an exact integer isn’t possible.
• Simplify algebraic manipulations and provide exact forms.

Mastering radicals becomes especially useful in higher mathematics where solutions often require exact values rather than decimal approximations.