Problem 13 - Introductory Algebra for College Students

Question

Solve each quadratic equation by the square root property. If possible, simplify radicals or rationalize denominators. $$2 x^{2}+1=51$$

Step-by-Step Solution

Answer

The short answer is \(x = 5, -5\). The solutions for the equation are 5 and -5.

Step 1: Rearrange the Equation

The equation given is \(2x^{2} + 1 = 51\). Start off by subtracting 1 from both sides of the equation. This gives the rearranged equation \(2x^{2} = 50\).

Step 2: Isolate \(x^{2}\) term

To isolate the \(x^{2}\) term, divide the entire equation by 2, hence: \(x^{2} = 25\).

Step 3: Apply Square Root Property

Applying the square root property implies that if \(x^{2} = a\), then \(x = ±\sqrt{a}\). In this case, \(x^{2} = 25\), therefore \(x = ±\sqrt{25}\).

Step 4: Simplify the Result

The square root of 25 is 5, hence the solutions of the equation are \(x = 5\) and \(x = -5\).

Key Concepts

Square Root Property

The Square Root Property is a key technique for solving quadratic equations of the form \( x^2 = a \). This property states that if \( x^2 = a \), then the solutions for \( x \) are \( x = \pm \sqrt{a} \). This means there are typically two solutions: one positive and one negative. This is especially useful when a quadratic equation can be restructured so its left-hand side becomes a perfect square.
In practice, you first rearrange the given quadratic equation to get \( x^2 \) by itself on one side. For instance, if you have an equation like \( 2x^2 + 1 = 51 \) as in the original exercise, rearranging steps help you isolate the \( x^2 \) term, eventually leading to \( x^2 = 25 \).
Once the equation is in this form, apply the Square Root Property to find \( x = \pm 5 \). Remember:

• Ensure the \( x^2 \) term is isolated on one side.
• Use the \( \pm \) symbol to capture both possible solutions.

This property simplifies the process and bypasses more complex factoring or completing the square methods.

Simplifying Radicals

Simplifying radicals involves making the expression under the square root sign as simple as possible. In its simplest terms, it requires expressing radicals in their simplest form. For example, \( \sqrt{25} \) is already simplified because 25 is a perfect square.
Here's how you can simplify radicals:

• Break down the number or expression into its prime factors.
• Look for pairs of factors because these can be "taken out" of the radical sign. For example, \( \sqrt{36} = \sqrt{6 \times 6} = 6 \).
• If the number isn’t a perfect square, look for the largest perfect square factor. For instance, \( \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \).

In the context of quadratic equations, simplifying radicals often directly follows applying the Square Root Property. This step ensures that the solutions appear in their cleanest form, which can make recognizing further properties or solutions easier.

Rationalizing Denominators

Rationalizing the denominator is a method used to eliminate radicals from the denominator of a fraction. It makes it easier to deal with numbers and simplifies the expression for further calculations.
Here's the basic way to rationalize denominators:

• If you have a fraction like \( \frac{a}{\sqrt{b}} \), multiply both the numerator and the denominator by \( \sqrt{b} \) to eliminate the radical in the denominator: \( \frac{a}{\sqrt{b}} = \frac{a \sqrt{b}}{b} \).
• For more complex expressions, like \( \frac{a}{\sqrt{b} + c} \), multiplying by the conjugate \( \sqrt{b} - c \) and simplifying the result will help rationalize the fraction.

Although not directly applied in every quadratic equation, rationalizing the denominator is critical when dealing with solutions that include fractions or radical terms, ensuring that the final answer is clean and easy to understand.