Problem 4 - 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. $$(3 x-2)^{2}=2$$

Step-by-Step Solution

Answer

Yes, it is quadratic with \( a = 9 \), \( b = -12 \), \( c = 2 \).

Step 1: Understand the Form of Quadratic Equation

A quadratic equation is typically in the form \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). Our task is to see if the given equation can be rearranged into this form.

Step 2: Expand the Left Side

Start by expanding the left side of the given equation: \[ (3x - 2)^2 = (3x - 2)(3x - 2) = 9x^2 - 12x + 4 \].This uses the binomial expansion formula \( (a-b)^2 = a^2 - 2ab + b^2 \).

Step 3: Write in a Quadratic Form

Substitute the expansion back into the equation to get:\[ 9x^2 - 12x + 4 = 2 \].Now we will simplify the equation to have it in the form \( ax^2 + bx + c = 0 \).

Step 4: Rearrange to Standard Quadratic Form

Subtract 2 from both sides to move the constant term to the left side:\[ 9x^2 - 12x + 4 - 2 = 0 \],which simplifies to:\[ 9x^2 - 12x + 2 = 0 \].This equation is now in the standard quadratic form.

Step 5: Identify the Coefficients

Compare the equation \( 9x^2 - 12x + 2 = 0 \) with the standard form \( ax^2 + bx + c = 0 \).Identify the coefficients: \( a = 9 \), \( b = -12 \), and \( c = 2 \). Since \( a > 0 \), the equation is quadratic.

Key Concepts

Binomial Expansion

To determine if an equation is quadratic, it can be helpful to expand any squared terms, particularly those involving binomials. A binomial expansion involves breaking down expressions raised to a power greater than one. For instance, when given the binomial expression \((3x - 2)^2\), you can use the expansion formula: \((a-b)^2 = a^2 - 2ab + b^2\). By applying this, we take each component:

• Square the first term: \(a^2 = (3x)^2 = 9x^2\)
• Multiply the two terms together and double it: \(-2ab = -2 \times 3x \times 2 = -12x\)
• Square the second term: \(b^2 = (-2)^2 = 4\)

Combining these results gives us \(9x^2 - 12x + 4\). This expanded form allows us to see the quadratic components more clearly, confirming that the expression involves quadratic terms.

Coefficient Identification

Identifying coefficients in a polynomial is key to understanding its structure, particularly when verifying if it is a quadratic equation. Once an equation has been expanded and rearranged into the form \(ax^2 + bx + c = 0\), we can look at each component:

• The coefficient \(a\) is the number in front of \(x^2\). Here, \(a = 9\).
• The coefficient \(b\) is associated with \(x\). For this equation, \(b = -12\).
• Lastly, the constant \(c\) is the standalone number, which is \(2\) in this example.

Recognizing these coefficients not only confirms the equation's quadratic nature but also sets the stage for solutions and graph interpretations. Critical to this identification is ensuring \(a eq 0\), because if \(a\) were zero, the equation would no longer be quadratic.

Standard Quadratic Form

The standard quadratic form \( ax^2 + bx + c = 0 \) is the baseline for analyzing and solving quadratic equations. To rearrange an equation into this form, isolate all terms on one side with zero on the other, ensuring they fit the pattern:

• Start by expanding and simplifying any complex terms, such as those acquired through binomial expansion.
• Then, rearrange terms to match \(ax^2 + bx + c = 0\).

In our example, the equation simplifies to \(9x^2 - 12x + 4 = 2\). Subtracting 2 from both sides gives \(9x^2 - 12x + 2 = 0\). This step is crucial, as achieving this form confirms the equation is quadratic, allowing us to identify \(a\), \(b\), and \(c\) and qualify the nature of the equation (e.g., number of solutions, type of graph). It serves as a necessary step before employing further methods like factoring, completing the square, or using the quadratic formula to find solutions.