Problem 34 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ x^{2}-100=0 $$

Step-by-Step Solution

Answer

x = \pm 10

Step 1: Isolate the Quadratic Term

First, isolate the quadratic term by adding 100 to both sides of the equation:\[ x^2 - 100 + 100 = 0 + 100 \]This simplifies to:\[ x^2 = 100 \]

Step 2: Apply the Square Root Property

To solve for \( x \), take the square root of both sides of the equation. Remember to include both the positive and negative roots:\[ x = \pm \sqrt{100} \]

Step 3: Simplify the Radicals

Simplify the square root of 100:\[ x = \pm 10 \]

Key Concepts

Square Root Property

The square root property is an essential tool in solving quadratic equations. It states that if you have an equation of the form \[ x^2 = a \] then the solution is \[ x = \textpm \sqrt{a} \] This property is particularly useful when the quadratic term is isolated, meaning it stands alone on one side of the equation. Using the square root property helps us to find both the positive and negative roots of the quadratic equation. This means solving \[ x^2 = 100 \] by taking the square root of both sides to get \[ x = \textpm 10 \] .

Isolating the Quadratic Term

Before applying the square root property, the quadratic term must be isolated. This involves moving all other terms to the opposite side of the equation. In our example, the original equation is \[ x^2 - 100 = 0 \] To isolate \( x^2 \), add 100 to both sides: \[ x^2 - 100 + 100 = 0 + 100 \] This simplifies to: \[ x^2 = 100 \] Now, the quadratic term is isolated, and you can proceed with solving the equation using the square root property.

Simplifying Radicals

After applying the square root property, you often end up with a radical expression. Simplifying radicals is the process of finding the simplest form of these expressions. Let's take \[ x = \textpm \sqrt{100} \] as an example. Since \( \sqrt{100} \) can be simplified to 10 (because 10 * 10 = 100), the simplest form of our solution is: \[ x = \textpm 10 \] It's important to always simplify radicals to their simplest form for clarity and accuracy.

Positive and Negative Roots

When solving quadratic equations, it's crucial to remember that taking the square root of both sides yields both a positive and a negative root. For instance, from \[ x^2 = 100 \] when we apply the square root property, we get: \[ x = \textpm 10 \] This means our solution includes both 10 and -10. Ignoring the negative root could lead to incomplete or incorrect answers. Hence, always consider \[ x = \textpm \sqrt{a} \] to ensure you catch both possible values of \( x \).