Problem 24 - College Algebra with Corequisite Support

Question

For the following exercises, solve the quadratic equation by using the square root property. $$ (x-5)^{2}=4 $$

Step-by-Step Solution

Answer

The solutions are \(x = 7\) and \(x = 3\).

Step 1: Recognize the Square Root Property

The square root property states that if you have an equation in the form \((x-a)^2 = b\), then \(x-a = \pm \sqrt{b}\). In this case, \((x-5)^2 = 4\), so we use the square root property.

Step 2: Apply the Square Root Property

We take the square root of both sides. By applying the square root property, we have: \(x-5 = \pm \sqrt{4}\).

Step 3: Simplify the Square Roots

The square root of 4 simplifies to 2. Therefore, the equation becomes: \(x-5 = \pm 2\).

Step 4: Solve for x by Adding 5

Now we solve for \(x\) by adding 5 to both sides for each case: 1. \(x-5 = 2\) gives \(x = 2+5 = 7\). 2. \(x-5 = -2\) gives \(x = -2+5 = 3\).

Step 5: Verify Solutions

Substitute \(x = 7\) and \(x = 3\) back into the equation \((x-5)^2 = 4\) to verify both solutions satisfy the original equation. For \(x = 7\), \((7-5)^2 = 2^2 = 4\). For \(x = 3\), \((3-5)^2 = (-2)^2 = 4\). Both solutions are verified.

Key Concepts

Square Root Property

The square root property is a straightforward method for solving specific quadratic equations. It is particularly useful when the equation has a perfect square on one side. The basic idea is that if a square of a number equals another number, then the original number is the positive or negative square root of the second number. This can be expressed in the formula: if \[(x - a)^2 = b\], then \[x - a = \pm \sqrt{b}\].
In practice, this means you can directly take the square root of both sides of the equation, remembering to consider both the positive and negative roots. For instance, with the equation \[(x - 5)^2 = 4\], applying the square root property gives:

• \(x - 5 = \sqrt{4}\)
• \(x - 5 = -\sqrt{4}\)

This results in two separate equations that you will solve individually to find the values of \(x\). The square root property simplifies the process, bypassing the need for traditional methods such as factoring or using the quadratic formula, especially when the equation is in the specific form needed.

Solving Quadratic Equations

Solving quadratic equations using the square root property often involves several straightforward steps. Once you've applied the square root property and simplified the square roots, the next step is solving for the variable \(x\). This process involves isolating \(x\) in the equations derived from the square root property.
Continuing with our example:

• \(x - 5 = 2\)
• \(x - 5 = -2\)

The next step is to solve each equation for \(x\) by performing arithmetic operations. In both cases, you simply move the constant term (which is -5 here) to the other side:

• For \(x - 5 = 2\): Add 5 to get \(x = 2 + 5 = 7\).
• For \(x - 5 = -2\): Add 5 to get \(x = -2 + 5 = 3\).

These operations provide us with the solutions for \(x\), namely \(x = 7\) and \(x = 3\), indicating that the parabola described by the equation intersects the x-axis at these points. This method is easy and efficient, especially for equations that neatly fit the square root property form.

Verifying Solutions

Verifying solutions is an essential final step to ensure the accuracy of your solutions. It confirms that the values you have found for \(x\) satisfy the original equation, preventing potential errors that could arise from arithmetic mistakes or incorrect application of the square root property.
To verify, substitute each solution back into the original equation and check if it holds true. For example, for our solutions \(x = 7\) and \(x = 3\), substitute back into the original equation:

• For \(x = 7\), substitute to get: \((7 - 5)^2 = 4\), which simplifies to \(2^2 = 4\).
• For \(x = 3\), substitute to get: \((3 - 5)^2 = 4\), which simplifies to \((-2)^2 = 4\).

In both cases, the left side equals the right side of the equation, confirming the solutions are correct. Verifying solutions finalizes the process, ensuring everything checks out before declaring the problem solved.