Problem 35 - Introductory Algebra for College Students

Question

Solve each quadratic equation by first factoring the perfect square trinomial on the left side. Then apply the square root property. Simplify radicals, if possible. $$x^{2}-10 x+25=2$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 5 + \sqrt{2}\) and \(x = 5 - \sqrt{2}\)

Step 1: Factoring Perfect Square Trinomial

The given equation is \(x^{2}-10 x+25 = 2\). Note that the left side of the equation is a perfect square trinomial. It can be factored as \((x-5)^2 \) as \(x^2 - 10x + 25 = (x-5)^2\). So the equation is now \((x-5)^2 = 2\)

Step 2: Applying the Square Root Property

Applying the square root property to both sides of the equation gives, \(x-5 = \sqrt{2}, -\sqrt{2}\). This is because if \(a^2 = b\), then \(a = \sqrt{b}, -\sqrt{b}\).

Step 3: Solving for X

Solving for \(x\) gives two possible values, \(x = 5 + \sqrt{2}\) and \(x = 5 - \sqrt{2}\)

Key Concepts

Factoring

Factoring is a key strategy when dealing with quadratic equations. It involves rewriting the equation in a form that is easier to solve. In a quadratic equation of the form \(ax^2 + bx + c\), we look for two numbers that multiply to \(ac\) and add to \(b\). However, when dealing with a perfect square trinomial, the factorization is even more straightforward.
For the equation \(x^2 - 10x + 25\), we can recognize that it's a perfect square trinomial. This means it can be expressed as the square of a binomial. Let's break it down step by step:

• Identify the perfect square trinomial: \(x^2 - 10x + 25\).
• Write it as a squared expression: \((x - 5)^2\).

Now, the equation \(x^2 - 10x + 25 = 2\) can be written as \((x - 5)^2 = 2\), which simplifies the solving process.

Perfect Square Trinomial

Understanding the concept of a perfect square trinomial is crucial in quadratics. A perfect square trinomial occurs when a quadratic can be expressed in the form \( (a + b)^2 \) or \( (a - b)^2 \). This results in the expanded form \( a^2 + 2ab + b^2 \) or \( a^2 - 2ab + b^2 \).
In the equation \(x^2 - 10x + 25\), each of the following expresses its properties:

• The first term \(x^2\) is a square of \(x\).
• The last term \(25\) is a square of \(5\).
• The middle term \(-10x\) corresponds to \(-2 \times x \times 5\).

This indicates that it is indeed a perfect square trinomial, represented as \((x - 5)^2\). Once you recognize this pattern, transforming the trinomial into its binomial squared form makes solving a breeze.

Square Root Property

The square root property is a handy tool to solve equations once you've rewritten them as a perfect square. It states that if \( a^2 = b \), then \( a = \pm \sqrt{b} \). This property allows for quick solutions of factored quadratic equations.
For our current equation \((x - 5)^2 = 2\), using the square root property means:

• Take the square root of both sides, giving \(x - 5 = \pm \sqrt{2} \).

This results in two linear equations:

• \(x - 5 = \sqrt{2}\)
• \(x - 5 = -\sqrt{2}\)

These can be easily solved to find \(x\), the solutions of the equation.

Simplifying Radicals

Simplifying radicals is about making square roots and other radical expressions as simple as possible. For the equation we're working with, after applying the square root property, we have terms \(\sqrt{2}\) and \(-\sqrt{2}\).
Some basic rules when simplifying radicals include:

• Ensure there are no perfect squares within the radical (i.e., \(\sqrt{2}\) is already simplified).
• Make sure the fraction within a radical is in its simplest form if applicable.

In our problem, \(\sqrt{2}\) is already in its simplest form because 2 isn't divisible by any perfect square numbers other than 1.
Expressing the solutions as \(x = 5 + \sqrt{2}\) and \(x = 5 - \sqrt{2}\) is both clear and concise, providing simplified results to the quadratic equation. This way, the solutions are ready to be applied to any further calculations or analyses.