Problem 7 - Introductory Algebra for College Students

Question

Solve each quadratic equation by the square root property. If possible, simplify radicals or rationalize denominators. $$x^{2}=50$$

Step-by-Step Solution

Answer

The solution to the equation \(x^2 = 50\) is \(x = 5\sqrt{2}\) and \(x = -5\sqrt{2}\).

Step 1: Identify the Value

Identify the value to take square root of. In terms of the given equation, we have \(x^2 = 50\). So, our value for square root is \(50\).

Step 2: Apply Square Root Property

Apply square root property to find the value of \(x\). By doing so, we get two possible solutions: \(x = \sqrt{50}\) and \(x = -\sqrt{50}\). Being a quadratic equation, it must have two solutions.

Step 3: Simplify the Radicals

Simplify the radicals. In this case, square root of \(50\) can also be written as \(\sqrt{25} \times \sqrt{2}\), which simplifies to \(5\sqrt{2}\). So, the solutions become \(x = 5\sqrt{2}\) and \(x = -5\sqrt{2}\).

Key Concepts

Square Root Property

The square root property is a straightforward way to solve quadratic equations that do not have both an \(x\) term and a constant. It is particularly useful when the equation is in the form \(x^2 = c\), where \(c\) is a non-negative constant. Applying this property involves:

• Taking the square root on both sides of the equation.
• Remembering that taking the square root of a number gives both its positive and negative roots.

For example, if we have \(x^2 = 50\), applying the square root property would give us \(x = \sqrt{50}\) and \(x = -\sqrt{50}\). This is because both \((\sqrt{50})^2\) and \((-\sqrt{50})^2\) would yield 50.
Using this property allows you to find potential solutions quickly, especially when the quadratic equation doesn't contain a linear term.

Simplify Radicals

Simplifying radicals is an important step in solving equations involving roots. It allows us to express the roots in their simplest form for ease of calculation and better understanding. Here's how to simplify radicals:

• Identify factors of the radicand (the number inside the radical) that are perfect squares.
• Break down the radicand into these factors.
• Take the square root of the perfect square factors.

In our example, simplifying \(\sqrt{50}\) involves recognizing that 50 can be factored into 25 and 2. As such:

• \(\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2}\)
• Since \(\sqrt{25} = 5\), \(\sqrt{50}\) simplifies to \(5\sqrt{2}\).

Simplifying radicals not only helps in providing more exact solutions but also often simplifies further mathematical operations with the solutions.

Rationalize Denominators

Rationalizing denominators is a process not directly needed for the initial problem, but it is a crucial technique in algebra when dealing with roots in denominators. This process makes expressions easier to handle by removing radicals from the denominator.
Here's how to do it:

• Multiply the numerator and the denominator by the conjugate of the denominator if it’s a binomial.
• For a single-term denominator that is a square root, multiply the numerator and denominator by the same root.

This method works because multiplying a radical by itself eliminates the radical in the denominator. For instance, if you have \(\frac{1}{\sqrt{2}}\), you would multiply by \(\frac{\sqrt{2}}{\sqrt{2}}\) to get \(\frac{\sqrt{2}}{2}\).
Though not immediately necessary for solving our specific quadratic equation, understanding rationalizing denominators is immensely helpful for complex problems involving fractions with roots.