Problem 19 - College Algebra and Calculus: An Applied Approach

Question

Use the Quadratic Formula to solve the quadratic equation. $$ 12 x-9 x^{2}=-3 $$

Step-by-Step Solution

Answer

The equation \(12x - 9x^2 = -3\) has solutions \(x=\frac{1}{2}, \frac{2}{3}\).

Step 1: Rearrange the equation to match standard quadratic form

Rearrange the given quadratic expression \(12x - 9x^2 = -3\) to the standard quadratic form. This can be achieved by applying the additive inverse of each side of the equation, resulting in \(9x^2 - 12x + 3 = 0\).

Step 2: Identify coefficients

In the rearranged equation \(9x^2 - 12x + 3 = 0\), identify the coefficients for \(a\), \(b\), and \(c\). In this equation, \(a = 9\), \(b = -12\), and \(c = 3\).

Step 3: Use the Quadratic Formula

Apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substituting the identified coefficients into the formula, we get \(x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4*9*3}}{2*9}\).

Step 4: Simplify the solutions

Simplify the terms under and outside of the square root to get the two possible solutions. This gives \(x = \frac{12 \pm \sqrt{144 - 108}}{18}\), which simplifies to \(x = \frac{12 \pm \sqrt{36}}{18}\), yielding final solutions of \(x=\frac{1}{2}, \frac{2}{3}\).

Key Concepts

Quadratic Equations

Quadratic equations can seem complicated at first glance, but they are just a special type of equation. They involve variables raised to the power of two. Specifically, they can be written in the format \( ax^2 + bx + c = 0 \). The main goal in working with quadratic equations is often to find the values of \( x \) that make the equation true—these are called the "solutions" or "roots." Quadratic equations pop up in various real-world scenarios, such as calculating areas, understanding the motion of objects, or even optimizing business profits. Understanding how to solve them is crucial and also gives you tools to tackle more complex mathematical challenges later on.

Standard Quadratic Form

The term "standard quadratic form" might sound a bit intimidating, but it's simply a way to write a quadratic equation in a standard format as \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are known as coefficients, and they are typically numbers.
The purpose of having a standard form is to simplify the process of identifying and solving the equation. Any quadratic equation, like the example given, can be rearranged into this form. Rearranging involves ensuring that all terms are on one side of the equation, set to equal zero. This uniform approach makes it easier to apply methods like factoring, completing the square, or using the quadratic formula to find the solutions.

Coefficients Identification

Once your quadratic equation is in standard form, the next step is identifying the coefficients \( a \), \( b \), and \( c \). These coefficients play a key role in determining the nature and number of solutions for the equation.
In the example equation \( 9x^2 - 12x + 3 = 0 \), the coefficients are as follows:

• \( a = 9 \) - This is the coefficient of \( x^2 \).
• \( b = -12 \) - This is the coefficient of \( x \).
• \( c = 3 \) - This is the constant term.

Identifying these correctly is crucial as they will be used in the quadratic formula to find the solutions to the equation.

Solving Quadratic Equations

To solve quadratic equations, one effective method is using the quadratic formula. This formula is a direct way to find the solutions (or roots) of any quadratic equation, provided you have the coefficients identified.
The quadratic formula is given as:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

• Substitute the coefficients into the formula.
• Calculate the value inside the square root, known as the discriminant \( b^2 - 4ac \). The discriminant tells us how many real solutions exist.
• Solve for \( x \) using the positive and negative square root values.

In our example, substituting \( a = 9 \), \( b = -12 \), and \( c = 3 \), the formula provides two solutions: \( x = \frac{1}{2} \) and \( x = \frac{2}{3} \). Each solution is a value where the original quadratic equation equals zero.