Problem 6 - College Algebra and Calculus: An Applied Approach

Question

Write the quadratic equation in general form. $$ 12-3(x+7)^{2}=0 $$

Step-by-Step Solution

Answer

The quadratic equation in general form is \(x^2 + 14x + 45 = 0\).

Step 1: Expansion

Expand \( (x+7)^{2}\). This expansion will give \(x^2+14x+49\).

Step 2: Distribute the negative

Distribute the -3 in front of \((x+7)^2\). This gives \(-3x^2-42x-147\).

Step 3: Rearrange equation

Rearrange \(12 - 3(x+7)^2 = 0\) into the general quadratic form. Now we have \(-3x^2-42x-135=0\).

Step 4: Simplify equation

Simplify equation by dividing every term by -3. This gives: \(x^2 + 14x + 45 =0\).

Key Concepts

Algebra

Algebra is a fundamental branch of mathematics that involves using symbols and letters to represent numbers and operations. It forms the basis for expressing and solving equations. When dealing with algebra, it's important to understand how to manipulate these symbols to find unknown values.
For example, in the equation \( 12 - 3(x+7)^{2} = 0 \), we suspect that it represents a standard mathematical relationship, and algebra helps us explore this relationship.
We apply algebraic operations systematically to transform the equation into a more familiar form, making it easier to identify solutions.

Polynomial Expansion

Polynomial expansion is a technique used to express a mathematical expression in its simplest form. In our exercise, we begin by expanding the term \((x+7)^{2}\).
Expansion means to multiply the terms inside the parentheses by themselves. So, \((x+7)(x+7)\) becomes:

• \(x^2\) – from \(x\times x\)
• \(7x\) – from \(x\times 7\)
• Another \(7x\) – from \(7\times x\)
• \(49\) – from \(7\times 7\)

By summing these, you get \(x^2 + 14x + 49\). This clear breakdown of polynomial components helps simplify complex expressions into manageable terms.

Quadratic Formula

The quadratic formula is a tool for finding the solutions to quadratic equations. A quadratic equation is generally expressed in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants.
The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula offers a straightforward way to determine the values of \(x\) that satisfy the equation by merely substituting the equation’s coefficients into the formula.
Understanding this formula is crucial because it guarantees solutions for any quadratic equation, even when factoring isn’t feasible.

Equation Simplification

Equation simplification aims to make an equation as straightforward as possible. In the exercise, after expanding and rearranging the terms, we found ourselves with the equation \(-3x^2-42x-135=0\).
To simplify, each term of the equation is divided by \(-3\). Simplifying reduces \(-3x^2-42x-135\) to \(x^2+14x+45\).
This process is crucial for making equations easier to understand and solve, as it brings the equation to its most fundamental form, often revealing insights or solutions that are less apparent in more complex forms.