Problem 38 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ x^{2}-8=0 $$

Step-by-Step Solution

Answer

x = \pm 2\sqrt{2}

Step 1: Isolate the quadratic term

First, add 8 to both sides of the equation to isolate the quadratic term. \[ x^{2} - 8 + 8 = 0 + 8 \] This simplifies to: \[ x^{2} = 8 \]

Step 2: Apply the square root property

Next, take the square root of both sides of the equation. Remember to include both the positive and negative roots. \[ x = \pm \sqrt{8} \]

Step 3: Simplify the radical

Simplify the expression under the radical sign. \[ \sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \] So, \[ x = \pm 2\sqrt{2} \]

Key Concepts

square root property

The square root property is a valuable tool for solving quadratic equations, especially when the quadratic term is isolated. Here's how it works. If you have an equation in the form of \[x^{2} = k\]you can solve for \(x\) by taking the square root of both sides. This will give you two solutions: one positive and one negative.

For example, with our problem, the equation is \[x^{2} = 8\] Applying the square root property, we take the square root of both sides: \[x = \pm \sqrt{8}\]
This results in two potential solutions. Whenever you are taking the square root of a number, do not forget to consider both the positive and negative roots.

simplifying radicals

Simplifying radicals involves reducing the radical expression to its simplest form. Let's break it down step-by-step.

Looking at our problem, we have: \(\sqrt{8}\)
To simplify it, we need to find the prime factors of 8. Since \(8 = 4 \cdot 2\)we can rewrite this as: \(\sqrt{8} = \sqrt{4 \cdot 2}\)
Next, apply the property of square roots that allows us to separate the product under the radical: \(\sqrt{4 \cdot 2} = \sqrt{4}\cdot\sqrt{2}\)
Since \(\sqrt{4} = 2\), we get: \(\sqrt{8} = 2\sqrt{2}\)
This simplification indicates that: \(x = \pm2\sqrt{2}\)
Always look to find perfect squares to simplify the radicals as much as possible.

steps in solving equations

Solving equations, particularly quadratic equations, involves a clear series of steps. Here's a simplified process you can follow to tackle these problems efficiently.

• Step 1: Isolate the quadratic term. Start by moving all other terms to the opposite side of the equation. For example, in \(x^{2} - 8 = 0\), add 8 to both sides to get \(x^{2} = 8\).
• Step 2: Apply the square root property. Once the quadratic term is isolated, take the square root of both sides, remembering to include both the positive and negative roots. This step transforms \(x^{2} = 8\) into \(x = \pm \sqrt{8}\).
• Step 3: Simplify radicals. Break down any radical expressions into their simplest form. Here, \(\sqrt{8} = 2\sqrt{2}\). So, the final solution is \(x = \pm 2\sqrt{2}\).

By following these steps meticulously, you can confidently solve quadratic equations. Always review each step to ensure accuracy and completeness.