Problem 22 - College Algebra

Question

Solve the quadratic equation by using the square root property. $$ (x-3)^{2}=7 $$

Step-by-Step Solution

Answer

The solutions are \(x = 3 + \sqrt{7}\) and \(x = 3 - \sqrt{7}\).

Step 1: Understand the Square Root Property

The square root property states that if a square of a number is equal to a positive number, then that number is equal to the positive and negative square root of that number. We will apply this property to the equation \((x-3)^2=7\).

Step 2: Apply the Square Root Property

Take the square root of both sides of the equation. This gives us two possible equations. So, we write:\[x-3 = \sqrt{7}\quad \text{or}\quad x-3 = -\sqrt{7}\]

Step 3: Solve for x in the First Equation

Solve the equation \(x-3 = \sqrt{7}\) to find one solution for \(x\):\[x = 3 + \sqrt{7}\]

Step 4: Solve for x in the Second Equation

Solve the equation \(x-3 = -\sqrt{7}\) to find the other solution for \(x\):\[x = 3 - \sqrt{7}\]

Step 5: Conclude the Solution

The solutions to the equation \((x-3)^2 = 7\) are \(x = 3 + \sqrt{7}\) and \(x = 3 - \sqrt{7}\).

Key Concepts

Square Root Property

The square root property is a handy tool for solving quadratic equations, especially when dealing with equations in the form of \((x-a)^2 = b\). The property states that if you have a squared term equal to a positive number, the quantity inside the square can be equal to either the positive or negative square root of that number.By applying this property, you're effectively splitting the equation into two simpler equations. Let's look at how it works in a practical example:

• If \((x-3)^2 = 7\), applying the square root property means: \[x-3 = \pm \sqrt{7}\]
• This splits into: \[x-3 = \sqrt{7}\] or \[x-3 = -\sqrt{7}\]

By understanding this concept, you can solve many quadratic equations efficiently without needing to expand or rearrange complex equations. Once you take the square root, simple algebraic steps will lead you to the solution.

Solving Equations

Once the square root property is applied, solving the resulting equations becomes straightforward. You solve for the variable by isolating it on one side of the equation.
Here's a step-by-step guide:

• Start with one of the equations: \[x - 3 = \sqrt{7}\]
• Add 3 to both sides to isolate \(x\): \[x = 3 + \sqrt{7}\]

Repeat the process for the other equation:

• Start with: \[x - 3 = -\sqrt{7}\]
• Add 3 to both sides: \[x = 3 - \sqrt{7}\]

By solving the two equations, we find two possible values for \(x\). This illustrates an important aspect of quadratic equations—they often have two solutions due to their nature.

Algebraic Solutions

In algebra, finding solutions to equations means determining the unknown variable's value that satisfies the equation. In the case of the equation \((x-3)^2 = 7\), through algebraic manipulation and reasoning, we find two valid solutions.
The algebraic process involves simplifying expressions and isolating variables. The basic operations—addition, subtraction, multiplication, and division—are used extensively. For quadratic equations:

• Recognize the structure: Learn to identify equations where the square root property can be efficiently applied, such as \((x-a)^2\).
• Isolate the variable: Once split into two simpler equations, use algebra to solve for the variable \(x\).
• Check your solutions: Substitute back into the original equation if necessary to ensure both solutions are correct and represent actual solutions to the equation.

Algebraic solutions provide a powerful way to deal with mathematical problems by using logical processes to break down an equation into more manageable parts.