Problem 22 - College Algebra with Corequisite Support

Question

For the following exercises, 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: Identify the Quadratic Equation

The given quadratic equation is \((x-3)^2 = 7\). This equation is already in the form suitable for using the square root property, as it involves a perfect square on one side.

Step 2: Apply the Square Root Property

According to the square root property, if \(a^2 = b\), then \(a = \pm \sqrt{b}\). Applying this to \((x-3)^2 = 7\), we take the square root of both sides: \[x - 3 = \pm \sqrt{7}\]

Step 3: Solve for x

Next, we solve for \(x\) by isolating it. Add 3 to both sides to obtain:\[x = 3 \pm \sqrt{7}\]

Step 4: Present the Solutions

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 powerful tool to solve certain types of quadratic equations. It applies when you have an equation with a perfect square on one side.

• For example, if you have \(a^2 = b\), then using the square root property means \(a = \pm \sqrt{b}\). It helps simplify and solve the equation by turning a square into its roots.
• It’s crucial to remember the \(+\) and \(-\) solutions because squaring real numbers always results in positive values.
• In our exercise, we started with \( (x-3)^2 = 7\), which is perfectly set up for this method.

By using the square root property, you can reduce the complexity of the equation and find your solutions easily: \(x - 3 = \pm \sqrt{7}\). This brings us closer to isolating the variable, leading to the final solutions.

Solving Quadratics

Solving quadratic equations can be achieved through various methods, including factoring, completing the square, or using the quadratic formula. For equations like the one given in the task, we used the square root property.

• First, ensure that the quadratic is in the form where one side is a perfect square.
• Next, apply the square root property to simplify the equation.
• Finally, solve for the variable by performing basic algebraic operations.

In the task, once we had \(x - 3 = \pm \sqrt{7}\), all required was adding 3 to both sides: \(x = 3 \pm \sqrt{7}\). This straightforward logical flow is why the square root property is a favored method for suitable quadratics.

Perfect Squares

A perfect square is an expression squared, typically of the form \( (a-b)^2 \). Identifying these can greatly ease solving quadratics.

• Perfect squares are key to using the square root property effectively.
• They let us express complex equations as simple roots.
• Recognizing \( (x-3)^2 \) as a perfect square aligned us with a direct approach for solving.

They simplify and accelerate the solution process. For example, rewriting an equation to expose its perfect square form helps reduce ambiguity in solving.