Problem 26 - Algebra for College Students

Question

Use the method of completing the square to solve each quadratic equation. $$n^{2}+n-1=0$$

Step-by-Step Solution

Answer

The solutions are \(n = \frac{-1 + \sqrt{5}}{2}\) and \(n = \frac{-1 - \sqrt{5}}{2}\).

Step 1: Write the equation in standard form

The equation is already in standard quadratic form: \(n^2 + n - 1 = 0\). In this form, the terms are \(an^2 + bn + c\) where \(a = 1\), \(b = 1\), and \(c = -1\).

Step 2: Move constant to the other side

Move the constant term \(-1\) to the right side of the equation: \(n^2 + n = 1\).

Step 3: Identify coefficients and compute necessary values

Identify the coefficient \(b=1\) and divide it by \(2\), computing \(\left(\frac{b}{2}\right)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}\).

Step 4: Add and subtract \(\left(\frac{b}{2}\right)^2\)

To complete the square, add \(\frac{1}{4}\) to both sides of the equation: \(n^2 + n + \frac{1}{4} = 1 + \frac{1}{4}\).

Step 5: Simplify the right side

Add \(1\) and \(\frac{1}{4}\) together: \(1 + \frac{1}{4} = \frac{5}{4}\). Now the equation becomes \(n^2 + n + \frac{1}{4} = \frac{5}{4}\).

Step 6: Factor the left side into a perfect square

The left side \(n^2 + n + \frac{1}{4}\) is a perfect square trinomial and can be written as \(\left(n + \frac{1}{2}\right)^2\). The equation now is \(\left(n + \frac{1}{2}\right)^2 = \frac{5}{4}\).

Step 7: Take the square root of both sides

Take the square root of both sides: \(n + \frac{1}{2} = \pm\sqrt{\frac{5}{4}}\). This simplifies to \(n + \frac{1}{2} = \pm\frac{\sqrt{5}}{2}\).

Step 8: Solve for \(n\)

Subtract \(\frac{1}{2}\) from both sides. This gives two solutions: \(n = -\frac{1}{2} + \frac{\sqrt{5}}{2}\) or \(n = -\frac{1}{2} - \frac{\sqrt{5}}{2}\). Simplify these expressions if necessary.

Key Concepts

Quadratic Equations

A quadratic equation is a type of polynomial equation where the highest degree of the unknown variable is 2. This means it takes on the general form of \(an^2 + bn + c = 0\). In this expression:

• \(a\), \(b\), and \(c\) are constants, where \(a eq 0\).
• \(n\) is the variable.

Quadratic equations can appear in various scenarios, from physics problems describing parabolic trajectories to any situation involving square-shaped patterns. In our given problem, the quadratic equation is \(n^2 + n - 1 = 0\). Here, \(a = 1\), \(b = 1\), and \(c = -1\).
To solve a quadratic equation such as this one, several methods can be used, including factoring, using the quadratic formula, or completing the square. Each method offers unique advantages, depending on the specific characteristics of the equation at hand.

Perfect Square Trinomial

A perfect square trinomial is a special form of quadratic expression. It's called "perfect square" because it can be expressed as the square of a binomial. The form generally looks like \((a + b)^2\), which expands to \(a^2 + 2ab + b^2\).
Understanding this form is crucial when completing the square. The task involves manipulating a given quadratic expression so that it takes on this squared binomial form. In our example, \(n^2 + n + \frac{1}{4}\) becomes \((n + \frac{1}{2})^2\). This transformation is useful since it allows for direct application of the square root property to solve the equation.
To identify a perfect square trinomial when completing the square:

• Take the coefficient of the linear term \(b\),
• Divide it by 2,
• Square the result.

This result is added to both sides of the equation to maintain equality while transforming the original quadratic into a perfect square trinomial form.

Square Root Property

The square root property is a mathematical principle that allows us to solve equations involving squares. When you have an equation of the form \((n + p)^2 = q\), you can apply the square root property to simplify this to \(n + p = \pm \sqrt{q}\).
In our completed square equation \((n + \frac{1}{2})^2 = \frac{5}{4}\), the square root property is used to find \(n\). By taking the square root of both sides, we simply unravel the equation as follows:

• Take square roots: \(n + \frac{1}{2} = \pm \sqrt{\frac{5}{4}}\)
• Simplify \(\sqrt{\frac{5}{4}}\) to \(\frac{\sqrt{5}}{2}\)
• Subtract \(\frac{1}{2}\) from both sides, yielding the two solutions for \(n\).

This property is particularly handy because it provides a direct and efficient way to handle squares and extract their roots, simplifying the problem-solving process by turning a potentially complex quadratic equation into a straightforward arithmetic procedure.