Problem 33 - 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. $$x^{2}-6 x+9=36$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 9\) and \(x = -3\).

Step 1: Identify and Factor the Perfect Square Trinomial

The given equation is a perfect square trinomial, meaning it can be factored into \((x-a)^2\), where \(a\) is a constant. With the equation \(x^2 - 6x + 9\), it can be factored into \((x - 3)^2\). Thus, the equation now becomes \((x - 3)^2 = 36\).

Step 2: Apply the Square Root Property

Applying the square root property, which states that if \(a^2 = b\), then \(a\) will be equal to \(\sqrt{b}\) or \(-\sqrt{b}\). Hence, \(x - 3\) will be equal to \(\sqrt{36}\) or \(-\sqrt{36}\). Therefore, we have two equations: \(x - 3 = 6\) and \(x - 3 = -6\).

Step 3: Solve for x in Each Equation

Solving the first equation \(x - 3 = 6\) for \(x\), we get \(x = 6 + 3 = 9\). Solving the second equation \(x - 3 = -6\) for \(x\), we get \(x = -6 + 3 = -3\).

Key Concepts

Factoring Perfect Square Trinomials

Understanding how to factor perfect square trinomials is an essential skill in algebra, particularly when solving quadratic equations. A perfect square trinomial is a type of quadratic expression that results from squaring a binomial. In mathematical terms, it takes the form \(a^2 + 2ab + b^2\) and can be factored back into \(a + b)^2\) or \(a - b)^2\).

When you encounter an equation like \(x^2 - 6x + 9 = 36\), recognize that the first and last terms are squares of \(x\) and \(3\), respectively, and the middle term is twice the product of \(x\) and \(3\). This indicates that we're looking at a perfect square trinomial \(x^2 - 6x + 9\), which factors neatly into \(x - 3)^2\). Once factored, the equation is simpler to solve, as it reveals more about its structure and solutions.

Square Root Property

The square root property plays a pivotal role in solving quadratic equations, especially after factoring them into perfect squares. This property states that if \(a^2 = b\), then \(a = \sqrt{b}\) or \(a = -\sqrt{b}\), where \(\sqrt{b}\) denotes the principal (non-negative) square root of \(b\).

In our equation \( (x - 3)^2 = 36\), applying the square root property allows us to write two separate but related equations: \(x - 3 = \sqrt{36}\) and \(x - 3 = -\sqrt{36}\), which simplifies to \(x - 3 = 6\) and \(x - 3 = -6\), respectively. This step is crucial because it transforms a quadratic equation into two linear ones, making the process of finding the variable \(x\)'s value more straightforward.

Simplify Radicals

When working with square root property, it is often necessary to simplify radicals to make the solutions more apparent. Simplifying radicals involves expressing the square root of a number in its simplest form. For example, \(\sqrt{36}\) is simplified to \(6\) because \(36\) is a perfect square and \(6 \times 6 = 36\).

If the number under the radical weren't a perfect square, we would seek to express it as the product of a perfect square and another factor, then take the square root of the perfect square out of the radical. Simplifying radicals helps in understanding the exact values of the solutions and in confirming whether further simplification is needed. In our case, \(\sqrt{36}\) and \(\sqrt{-36}\) are already in their simplest forms, being \(6\) and \(\sqrt{-36}\) or \(6i\) (if considering complex solutions), respectively.