Problem 62 - Elementary Algebra

Question

\(n^{2}=10 n+8\)

Step-by-Step Solution

Answer

\(n = 5 + \sqrt{33}\) or \(n = 5 - \sqrt{33}\)

Step 1: Set the Equation to Zero

Rewrite the given equation by bringing all terms to one side to set it to zero: \(n^{2} - 10n - 8 = 0\)

Step 2: Identify Coefficients

Identify the coefficients from the quadratic equation in standard form \(an^{2} + bn + c = 0\). Here, \(a = 1\), \(b = -10\), and \(c = -8\).

Step 3: Use the Quadratic Formula

The quadratic formula is \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute the identified coefficients into this formula.

Step 4: Calculate the Discriminant

Calculate the discriminant, \(b^2 - 4ac\): \((-10)^2 - 4(1)(-8) = 100 + 32 = 132\).

Step 5: Compute the Solutions

Substitute the discriminant back into the quadratic formula to find the solutions for \(n\): \(n = \frac{10 \pm \sqrt{132}}{2} = \frac{10 \pm 2\sqrt{33}}{2} = 5 \pm \sqrt{33}\).

Key Concepts

Quadratic Formula

Quadratic equations can be solved using the quadratic formula, which is a powerful tool. It helps us find the values of the variable that make the equation true. The quadratic formula is: \[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here, the symbols have specific meanings:

• is the coefficient of \(n^2\)
• is the coefficient of \(n\)
• is the constant term

By plugging in these values from any quadratic equation, we can find the solutions, known as the roots of the equation. The formula derives from the process of completing the square and works for all quadratic equations.

Discriminant Calculation

The discriminant is a key part of the quadratic formula that helps determine the nature of the roots of the quadratic equation. It is calculated using: \[ b^2 - 4ac \]

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, the quadratic equation has exactly one real root, also known as a repeated or double root.
• If the discriminant is negative, the quadratic equation has two complex roots.

Through the discriminant, we can understand if the solutions are real or imaginary before actually calculating the roots.

Standard Form of Quadratic Equation

A quadratic equation is usually expressed in the standard form: \[ax^2 + bx + c = 0\]To use the quadratic formula effectively, the equation must be in this form. Here’s how to convert any given quadratic equation to standard form:

• Move all terms to one side of the equation
• Arrange them in descending order of their degrees

For example, for the equation \(n^2 = 10n + 8\), we subtract \(10n\) and 8 from both sides to get:\[n^2 - 10n - 8 = 0\]Now it’s in the standard form, ready for applying the quadratic formula to find the solutions.