Problem 34 - 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. $$x^{2}-6 x+9=49$$

Step-by-Step Solution

Answer

The solutions to the equation \(x^{2}-6 x+9=49\) are \(x=10\) and \(x=-4\).

Step 1: Factorize the perfect square trinomial

The left side of the equation \(x^{2}-6 x+9\) is a perfect square trinomial, which can be written in the form \((a-b)^2\). The equation becomes \((x-3)^2 = 49\).

Step 2: Apply the square root property

The square root property states: if \(x^2 = a\), then \(x = \sqrt{a}\) or \(x = -\sqrt{a}\). By applying this property, we get two possible solutions for \(x\), \(x = 3 + \sqrt{49}\) and \(x = 3 - \sqrt{49}\).

Step 3: Simplify radicals

Simplifying the radicals, we get \(x = 3 + 7 = 10\) and \(x = 3 - 7 = -4\)

Key Concepts

Factoring

Factoring is a fundamental skill in solving quadratic equations because it transforms a complex equation into a form that is easier to solve. In this exercise, we begin by recognizing a specific structure called a perfect square trinomial on the left-hand side: \(x^{2} - 6x + 9\). A perfect square trinomial is a quadratic expression that is the square of a binomial. The general form of a perfect square trinomial is \((a-b)^2 = a^2 - 2ab + b^2\). In this case, we observe that \(a = x\) and \(b = 3\), because \(2ab = 6x\) and \(b^2 = 9\). Thus, \(x^2 - 6x + 9 = (x-3)^2\). Recognizing these patterns quickly helps to factor more efficiently in problems like this one.

Perfect Square Trinomials

Perfect square trinomials are special quadratic expressions that form when a binomial is squared. The pattern \((a \, - \, b)^2 = a^2 - 2ab + b^2\) represents this. It means every perfect square trinomial can be expressed as the square of a binomial. In our equation, \(x^2 - 6x + 9\) turns into \((x - 3)^2\). This conversion is key because it simplifies our work as it allows us to focus on solving the equation in a structured way.

• Identify the binomial factors \(a\) and \(b\).
• Square to get the original trinomial.

The ease with which you can factor these trinomials once you understand the pattern makes problem-solving faster.

Square Root Property

The square root property is a technique used to solve equations where the variable is squared. Once we've factored our trinomial into \((x-3)^2 = 49\), the square root property allows us to solve for \(x\) by taking the square root of both sides of the equation. The property states: if \(x^2 = a\), then \(x = \sqrt{a}\) or \(x = -\sqrt{a}\). Applying this to our equation, we have two equations to solve:

• \(x - 3 = 7\)
• \(x - 3 = -7\)

Solving these gives us \(x = 10\) and \(x = -4\). This step is essential because it converts the problem into two simpler linear equations.

Simplifying Radicals

Simplifying radicals is the process of finding the simplest form of a square root. This is the final step in processes like this to make the solution more concise and clear. In our equation, after applying the square root property, we have \(\sqrt{49}\), which simplifies to \(7\) because \(49\) is a perfect square.

• Recognize and simplify perfect square roots.
• Simplified radicals help in providing neat solutions.

The act of simplifying radicals ensures that the solutions are as simplified as possible, leading to the final simplified answers of \(x = 10\) and \(x = -4\). Understanding this concept helps in maintaining accuracy and simplicity in the problem-solving process.