Problem 44 - Intermediate Algebra

Question

Simplify and reduce each expression. $$ n^{2}-4 n-192=0 $$

Step-by-Step Solution

Answer

The solutions are \( n = 16 \) and \( n = -12 \).

Step 1: Identify the equation type

The given expression is a quadratic equation in the form of \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -4 \), and \( c = -192 \). We will use the quadratic formula to find the roots.

Step 2: Apply Quadratic Formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Plugging in the identified values, \( a = 1 \), \( b = -4 \), and \( c = -192 \) into the formula, we get: \[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \times 1 \times (-192)}}{2 \times 1} \].

Step 3: Calculate the Discriminant

Calculate the discriminant \( b^2 - 4ac \). Substitute the values: \( (-4)^2 - 4 \times 1 \times (-192) = 16 + 768 = 784 \).

Step 4: Solve for Roots

Since the discriminant is 784, which is a perfect square (\( \sqrt{784} = 28 \)), calculate the roots: \[ x = \frac{4 \pm 28}{2} \].

Step 5: Compute Specific Values for x

Calculate the two possible values for \( x \): \( x = \frac{4 + 28}{2} = 16 \) and \( x = \frac{4 - 28}{2} = -12 \).

Step 6: Confirm Solutions

Both values, 16 and -12, are the solutions to the equation because \( (n - 16)(n + 12) = 0 \) tracks back to the original expression.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions to quadratic equations. A quadratic equation typically takes the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. The quadratic formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula allows us to determine the values of \( x \) that solve the equation. The "\( \pm \)" symbol indicates that there are generally two solutions, resulting from the addition and subtraction of the square root term.

Using the quadratic formula involves several straightforward steps:

• Identify the values of \( a \), \( b \), and \( c \) in the quadratic equation.
• Calculate the discriminant, which is \( b^2 - 4ac \).
• Substitute these values into the quadratic formula to find the roots.

This formula is extremely useful, particularly when the quadratic equation does not easily factorize.

Discriminant

The discriminant is a crucial part of the quadratic formula, represented as \( b^2 - 4ac \). It provides insight into the nature of the roots, or solutions, of the quadratic equation without solving it completely.

Understanding the discriminant:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, there is exactly one real root, often called a repeated or double root.
• If the discriminant is negative, the equation has two complex (non-real) roots.

In the given exercise, the discriminant is calculated as 784. Since this is a positive number and a perfect square, it indicates that the equation has two distinct real roots, which further simplifies our calculations.

Roots of a Quadratic Equation

The roots of a quadratic equation are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). These roots are essentially the solutions to the equation. Depending on the discriminant, the roots can be real or complex.

Here's how you can find the roots using the quadratic formula:

• When the discriminant is a perfect square, like in our exercise where it is 784, the roots are real and rational.
• The formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) gives us two solutions for \( x \) based on the "\( \pm \)" sign.
• Using the calculated discriminant, determine the precise values for \( x \).

In the above problem, the roots were calculated to be 16 and -12 by solving \( x = \frac{4 \pm 28}{2} \), illustrating that the equation roots are real numbers.