Problem 38 - 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. $$x^{2}+2 x+1=7$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x=-1 + \sqrt{7}\) and \(x=-1 - \sqrt{7}\).

Step 1: Identify and factor the perfect square trinomial

The given equation is \(x^{2}+2x+1=7\). From this, we have a perfect square trinomial, \(x^{2}+2x+1\), which factors into \((x+1)^{2}\).

Step 2: Apply the square root property

We rewrite the equation as \((x+1)^{2}=7\). Applying the square root property to both sides of the equation gives us \(x+1= \pm \sqrt{7}\).

Step 3: Solve for 'x'

To get 'x' alone, we subtract 1 from both sides of the equation. This gives \(x=-1 \pm \sqrt{7}\)

Key Concepts

Factoring Perfect Square Trinomials

Understanding how to factor perfect square trinomials is crucial when solving quadratic equations. A perfect square trinomial takes the form of \(a^2 + 2ab + b^2\) and can be factored into \(a + b)^2\). To recognize this pattern, look for a square of a term at both ends of the trinomial, and check if the middle term is twice the product of the two terms being squared.

When approaching an equation such as \(x^2 + 2x + 1 = 7\), we identify \(x^2\) as \(a^2\) and \(1\) as \(b^2\). The middle term, \(2x\), is indeed twice the product of \(x\) and \(1\), confirming that \(x^2 + 2x + 1\) is a perfect square trinomial. Factoring gives us \(x+1)^2\), which provides a much simpler expression to work with in the next steps of solving the quadratic equation.

Square Root Property

The square root property is an efficient way to solve equations where the variable part is squared, as is often the case with quadratic equations. According to this property, if you have an equation of the form \(a^2 = b\), you can take the square root of both sides to obtain \(a = \pm\sqrt{b}\). Remember, whenever you take the square root of a squared variable, there are always two possible solutions: a positive and a negative one.

In our exercise \(x^2 + 2x + 1 = 7\), after factoring the perfect square trinomial and rewriting the equation as \(\left(x+1\right)^2 = 7\), the square root property allows us to solve for \(x\) directly. By taking the square root of both sides, we obtain \(x + 1 = \pm\sqrt{7}\). This property helps us handle the squared term and leads us closer to finding the value of \(x\).

Simplifying Radicals

Simplifying radicals is a technique used to rewrite radical expressions in their simplest form. When you encounter a square root, look for factors of the radicand (the number under the root) that are perfect squares, as these can be taken out of the root in simplified form. For instance, \(\sqrt{18}\) can be simplified to \(3\sqrt{2}\) because \(18\) factors into \(9 \times 2\), where \(9\) is a perfect square.

In our equation, after applying the square root property, we are left with \(x = -1 \pm\sqrt{7}\). The radical \(\sqrt{7}\) cannot be simplified further since \(7\) has no perfect square factors other than \(1\). Thus, the solution to the equation is written in terms of a radical. Being comfortable with simplifying radicals will often lead to cleaner, more easily interpretable solutions—though in some cases, like this one, the simplest form of the radical is the radical itself.