Problem 19 - College Algebra

Question

Solve the quadratic equation by using the square root property. $$ x^{2}=36 $$

Step-by-Step Solution

Answer

The solutions are \(x = 6\) and \(x = -6\).

Step 1: Understand the Equation

The provided equation is a quadratic equation in the form \(x^2 = 36\). Our goal is to solve for \(x\) using the square root property.

Step 2: Apply the Square Root Property

The square root property states that if \(x^2 = a\), then \(x = \pm \sqrt{a}\). Here, \(a = 36\), so take the square root of both sides to get \(x = \pm \sqrt{36}\).

Step 3: Calculate the Square Roots

Calculate the square root of 36. The square root of 36 is \(6\). Therefore, the two possible solutions are \(x = 6\) and \(x = -6\).

Key Concepts

Square Root Property

The Square Root Property is a key method in algebra for solving quadratic equations, especially when they are expressed in a simple form. It is particularly useful when you have an equation like \(x^2 = a\). This property states that if \(x^2 = a\), then \(x = \pm \sqrt{a}\). Essentially, this means you take the square root of both sides of the equation to solve for \(x\).

Here's how it works in practice:

• Identify the term \(x^2 = a\) in your equation.
• Apply the square root to both sides, remembering to include both the positive and negative solutions, since squaring either will result in \(a\).

For example, if \(x^2 = 36\), you apply the property to find \(x = \pm \sqrt{36}\), giving you the solutions \(x = 6\) and \(x = -6\). This is straightforward when the right-hand side is a perfect square, but never forget to consider both the positive and negative square roots!

This method is a quick way to solve equations where one side is a perfect square.

Solving Equations

Solving equations is one of the fundamental skills in algebra. It involves finding values for variables that make an equation true. When dealing with quadratic equations, there can be multiple methods to find solutions.

It's essential to:

• Understand the type of equation: Quadratic equations are of the form \(ax^2 + bx + c = 0\).
• Choose the appropriate method: For equations in the form \(x^2 = a\), the square root property is efficient.
• Solve step-by-step: Break down the problem into manageable steps to avoid mistakes.

For the equation \(x^2 = 36\), the process can be simplified by just identifying \(a\), then applying the square root property as explained before.

Solving equations is about finding that balance and ensuring each step logically follows from the last, leading you towards the correct solution.

Algebraic Solutions

Algebraic solutions involve using algebraic manipulations to find the value of variables that satisfy an equation.

When solving quadratic equations using algebraic methods, such as the square root property, it's crucial to:

• Maintain equation balance: Whatever you do to one side of the equation, do the same to the other side.
• Account for all solutions: Quadratic equations often have two solutions due to the nature of squaring (positive and negative roots).
• Utilize algebraic identities and properties: These shortcut the solution process and ensure accuracy.

For instance, applying the square root property to \(x^2 = 36\) results in \(x = 6\) and \(x = -6\).

Algebraic solutions showcase the logical yet systematic aspect of algebraic manipulation, helping you arrive at the correct answer efficiently. This systematic approach ensures a deep understanding and accurate solution to equations.