Problem 12 - Algebra for College Students

Question

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$n(n-24)=-128$$

Step-by-Step Solution

Answer

The solutions are \(n = 8\) and \(n = 16\).

Step 1: Expand the Expression

Begin by expanding the expression on the left side of the equation. Multiply out the terms: \[n(n-24) = n^2 - 24n\]So, rewrite the equation as:\[n^2 - 24n = -128\]

Step 2: Move All Terms to One Side

Add 128 to both sides of the equation to set it to zero:\[n^2 - 24n + 128 = 0\]Now, the equation is set in a standard quadratic form: \(ax^2 + bx + c = 0\). The equation now looks like: \(n^2 - 24n + 128 = 0\).

Step 3: Factor the Quadratic Equation

We need to factor the quadratic equation \(n^2 - 24n + 128 = 0\). Look for two numbers whose product is 128 (the constant term, \(c\)) and whose sum is -24 (the coefficient of the linear term, \(b\)).The numbers -8 and -16 satisfy these conditions because:- \(-8 + (-16) = -24\)- \(-8 \times (-16) = 128\)Therefore, we can factor the equation as:\[(n - 8)(n - 16) = 0\]

Step 4: Solve the Factors Set to Zero

Set each factor to zero to find the possible values for \(n\):1. \(n - 8 = 0\) gives \(n = 8\)2. \(n - 16 = 0\) gives \(n = 16\)The solutions to the equation are \(n = 8\) and \(n = 16\).

Key Concepts

Quadratic Formula

When faced with a quadratic equation that isn’t easily factorable, the Quadratic Formula is a powerful tool. A quadratic equation takes the form \( ax^2 + bx + c = 0 \). The formula to find the roots of this equation is:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\]
The letters \( a \), \( b \), and \( c \) are constants that you identify from your specific equation:

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

By substituting these values into the formula, the possible solutions for \( x \) are revealed.
Remember that the part under the square root, \( b^2 - 4ac \), is called the discriminant, and it determines the nature of the roots:

• If the discriminant is positive, there are two real distinct solutions.
• If it is zero, there is exactly one real solution.
• If the discriminant is negative, the solutions are complex (non-real) numbers.

In problems where factoring is difficult, the quadratic formula simplifies finding the solutions.

Solving Quadratic Equations

To solve quadratic equations, you aim to find the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \).
There are several methods to solve these equations:

• Factoring the quadratic, like we practice in straightforward problems.
• Using the Quadratic Formula, especially when factorization is hard or impossible.
• Completing the square, which involves changing the equation into a perfect square form.
• Graphing, which involves plotting the equation and finding the points where it crosses the x-axis.

When solving by factoring, it requires finding two numbers that add up to give \( b \) and multiply to give \( ac \).
The solutions or "roots" can then be computed by setting each factor equal to zero, as each solution satisfies the original quadratic equation.

Polynomial Factoring

Factoring is a technique employed to simplify equations or expressions by expressing them as products of their factors. It's particularly useful in solving quadratic equations.
The goal in polynomial factoring is to break down the polynomial into simpler "factor" polynomials which, when multiplied, produce the original polynomial.
For quadratic equations, this typically involves expressions of the form \( (x - p)(x - q) \), where \( p \) and \( q \) are the roots of the equation.

• Identify two numbers that multiply to give the constant term \( c \).
• Ensure that they also sum up to the linear coefficient \( b \).
• These numbers help factor the expression into simpler terms that become evident roots.

In the provided solution, the quadratic equation \( n^2 - 24n + 128 = 0 \) was factored as \((n - 8)(n - 16) = 0\). Here, \(-8\) and \(-16\) are the numbers required for factoring, as seen by their sum and product aligning with the quadratic coefficients.
Reviewing factoring techniques helps reinforce understanding in broader algebraic contexts.