Problem 7 - Basic Technical Mathematics with Calculus

Question

Determine whether or not the given equations are quadratic. If the resulting form is quadratic, identify \(a, b,\) and \(c,\) with \(a>0 .\) Otherwise, explain why the resulting form is not quadratic. $$n\left(n^{2}+n-1\right)=n^{3}$$

Step-by-Step Solution

Answer

The equation is quadratic with coefficients: \(a = 1\), \(b = -1\), \(c = 0\).

Step 1: Distribute and Expand

First, distribute the term outside the parenthesis across the terms inside the parenthesis. For the equation \(n(n^2 + n - 1) = n^3\), perform the distribution to get \(n \cdot n^2 + n \cdot n - n \cdot 1 = n^3\). This simplifies to \(n^3 + n^2 - n = n^3\).

Step 2: Simplify the Equation

Next, simplify the equation by eliminating common terms from both sides. Subtract \(n^3\) from both sides to obtain \(n^2 - n = 0\).

Step 3: Verify if the Equation is Quadratic

A quadratic equation is in the form \(ax^2 + bx + c = 0\), where \(a eq 0\). Compare \(n^2 - n = 0\) with this form: \(a = 1\), \(b = -1\), \(c = 0\). Since \(a > 0\), this is a quadratic equation.

Step 4: Identify Coefficients

Now, identify the coefficients: \(a = 1\), \(b = -1\), and \(c = 0\). Ensure that \(a > 0\), which is true, confirming it as a quadratic equation.

Key Concepts

Quadratic Formula

When faced with a quadratic equation, one of the most powerful tools you can use is the quadratic formula. This nifty formula allows us to find the roots of any quadratic equation given in the form \( ax^2 + bx + c = 0 \). The formula is expressed as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

It's like your go-to gadget for solving these types of problems! This formula works by considering the coefficients \(a\), \(b\), and \(c\) which represent the numbers in front of \(x^2\), \(x\), and the constant term, respectively. By plugging these values into the quadratic formula, you can determine the values of \(x\) that satisfy the equation. For instance, if you identify \(a = 1\), \(b = -1\), and \(c = 0\), these become your data points to plug into the quadratic formula.

Coefficients in Quadratic Equations

Understanding coefficients is crucial when working with quadratic equations. In any expression of the form \(ax^2 + bx + c = 0\), each letter represents a coefficient or a constant. Here's how these components work together:

• \(a\) is the coefficient in front of \(x^2\). It must not be zero, as otherwise the equation wouldn't stay quadratic!
• \(b\) is the coefficient before \(x\). It influences the slope or linearity of the equation.
• \(c\) is the constant term, or simply speaking, it’s the plain number without an \(x\).

In our transformed equation \(n^2 - n = 0\), we identified the coefficients as \(a = 1\), \(b = -1\), and \(c = 0\). These tell us that the equation is quadratic, given that \(a > 0\). Recognizing and tailoring these coefficients is essential for writing or solving quadratic equations successfully.

Simplifying Equations

Simplifying equations is all about making things easier to work with. When you simplify, you rewrite the equation in a form that's most convenient to understand or solve. Here’s how you can do it:

• Start by distributing terms across parenthesis or brackets.
• Look for like terms on both sides of the equation and perform operations to keep the equation balanced.
• Eliminate identical terms from both sides of the equation whenever possible.

In the example \(n(n^2 + n - 1) = n^3\), we expanded it to \(n^3 + n^2 - n = n^3\), and further simplified by subtracting \(n^3\) from both sides, resulting in \(n^2 - n = 0\). Simplifying was key to realizing the problem fit within the quadratic equation framework. When solving equations, simplification is often a crucial first step, giving clarity and direction to proceed further with.