Problem 22 - Introductory Algebra for College Students

Question

Solve each quadratic equation by the square root property. If possible, simplify radicals or rationalize denominators. $$(x+6)^{2}=144$$

Step-by-Step Solution

Answer

The solutions to the given equation are \(x = 6\) and \(x = -18\)

Step 1: Applying the square root

The square root property states that if \((a)^{2}=b\), then \(a\) is equal to \(\sqrt{b}\) or \(-\sqrt{b}\). Applying this to our equation, \((x+6)^{2}=144\), gives \(x+6 =\sqrt{144}\) or \(x+6=-\sqrt{144}\).

Step 2: Simplifying the square root

\(\sqrt{144}\) is \(12\), so the equations now become \(x+6 =12\) or \(x+6=-12\).

Step 3: Solving for 'x'

Solving the first equation, \(x+6 =12\), for 'x' gives \(x=12-6=6\). Solving the second equation, \(x+6 =-12\), for 'x' gives \(x=-12-6=-18\).

Key Concepts

Square Root Property

The square root property is a handy technique for solving quadratic equations. It states that if you have an equation of the form \(a^2 = b\), you can find 'a' by taking the square root of both sides. So, \(a = \sqrt{b}\) or \(a = -\sqrt{b}\). This method is particularly useful when the equation can be easily rearranged into this format. In the given problem, \((x + 6)^2 = 144\), we can directly apply the square root property. By taking the square root of both sides, you simplify the equation allowing you to solve for the variable 'x.' This process breaks the equation into two separate solutions, giving you both possible values for 'x' that satisfy the original equation.

Simplify Radicals

Simplifying radicals often occurs after applying the square root property. A radical expression typically involves a square root, like \(\sqrt{144}\). To simplify it, you find the largest perfect square factor of the number under the radical. In our problem, \(\sqrt{144}\) simplifies to \(12\). Recognizing perfect squares quickly is key; it helps in transforming complex expressions into simpler ones. Another example is \(\sqrt{36}\), which simplifies to \(6\) since 36 is a perfect square. Simplifying radicals makes equations easier to work with and often leads to easier solutions for variables.

Rationalize Denominators

Rationalizing the denominator involves eliminating any radicals present in the denominator of a fraction. While not directly needed in our original exercise, it's a useful skill for making expressions look cleaner and more manageable. To rationalize, multiply both the numerator and denominator by a suitable radical that will cancel out the radical in the denominator. For example, to rationalize an expression like \(\frac{1}{\sqrt{3}}\), multiply both top and bottom by \(\sqrt{3}\), transforming it into \(\frac{\sqrt{3}}{3}\). This process helps when working with fractions in mathematical problems, ensuring all components are in a more simplified form.

Solving Quadratic Equations

Quadratic equations come in many forms, including \(ax^2 + bx + c = 0\). Solving them can involve different methods such as factoring, completing the square, or using the quadratic formula. In our specific exercise, we used the square root property which is effective for equations that are already or can be easily rearranged into the form \((x+p)^2 = q\). Each solution method involves isolating the variable and simplifying until the variable is alone on one side of the equation. For the given problem, after applying the square root property and simplifying, the solutions were straightforward to find, resulting in \(x = 6\) and \(x = -18\). Every method gives the equation context for solving efficiently depending on its structure.