Problem 36 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ x^{2}-13=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = \pm\sqrt{13} \).

Step 1: Isolate the Variable Term

Start by isolating the term that contains the variable. Add 13 to both sides of the equation:\[ x^{2} - 13 + 13 = 0 + 13 \] which simplifies to:\[ x^{2} = 13 \]

Step 2: Apply the Square Root Property

Take the square root of both sides of the equation to solve for \( x \). Remember to include both the positive and negative roots:\[ \sqrt{x^{2}} = \pm\sqrt{13} \] which simplifies to: \[ x = \pm\sqrt{13} \]

Step 3: Simplify the Radicals (if necessary)

In this case, \( \sqrt{13} \) is already in its simplest form because 13 is a prime number.

Key Concepts

Square Root Property

When solving quadratic equations, the square root property is a powerful tool. It allows us to directly solve for the variable by taking the square root of both sides of the equation. For example, if we have \( x^2 = a \), we can solve it by taking the square root of both sides, giving us \( x = \pm \sqrt{a} \). Remember, you always need to consider both the positive and negative roots when applying this property because squaring either a positive or negative number will result in the same value.

Isolating Variable

To effectively apply the square root property, we must first isolate the variable term. This step involves rearranging the equation so that the term containing the variable is by itself on one side of the equation. In the exercise \( x^2 - 13 = 0 \), we added 13 to both sides to isolate the \( x^2 \) term, resulting in the equation \( x^2 = 13 \). This ensures the equation is ready for the next steps.

• Start by adding, subtracting, multiplying, or dividing to isolate the variable term.
• Ensure the variable term has a positive coefficient when isolated.
• Make sure no other terms or constants are on the same side as the variable term.

Taking these steps correctly is crucial for accurately applying mathematical properties like the square root property.

Simplifying Radicals

Simplifying radicals means expressing the root in its simplest form. Radicals should have the smallest possible number inside the radical sign. For our problem \( x = \pm \sqrt{13} \), \( \sqrt{13} \) is already simplified because 13 is a prime number and cannot be broken down further. To simplify other radicals:

• Factor the number under the radical sign into its prime factors.
• Group the factors to pull out squares.
• Simplify the expression by taking the square root of the groups and pulling them out of the radical.

For example, if you had \( \sqrt{20} \), you would break it down to \( \sqrt{4 \cdot 5} \) and simplify to \( 2 \sqrt{5} \). This process ensures the expression is manageable and easier to work with.