Problem 19 - Introductory Algebra for College Students

Question

Solve each equation using the quadratic formula. Simplify irrational solutions, if possible. $$9 x^{2}-12 x+4=0$$

Step-by-Step Solution

Answer

The solution to the equation is \(x=\frac{2}{3}\)

Step 1: Identify the Coefficients

In the quadratic equation \(9x^{2}-12x+4=0\), the coefficient a is 9, b is -12, and c is 4.

Step 2: Substitute into the Quadratic Formula

Substitute the coefficients into the formula: \(x = \frac{--12 \pm \sqrt{(-12)^{2}-4*9*4}}{2*9}\), which simplifies to \(x = \frac{12 \pm \sqrt{144-144}}{18}\) .

Step 3: Simplify the Expression

Simplify the expression under the square root and the entire fraction: \(x = \frac{12 \pm \sqrt{0}}{18}\) . This further simplifies to \(x = \frac{12 \pm 0}{18}\), leading to two potential solutions: \(x = \frac{12}{18}\) or \(x = \frac{12}{18}\) . Both expressions simplify to \(x = \frac{2}{3}\), so there is only one solution.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). It's particularly useful when factoring the equation is difficult or impossible. The formula is as follows: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

• \( a \), \( b \), and \( c \) are the coefficients of the equation.
• The symbol \( \pm \) suggests that there can be two solutions.
• Because it involves a square root, the solutions may be real or complex numbers.

This formula is derived from the process of completing the square and is applicable to any quadratic equation.

Solving Equations

Solving quadratic equations involves finding the value(s) of \( x \) that make the equation true. Quadratic equations can have different types of solutions:

• Two distinct real solutions: If the expression under the square root, \( b^2 - 4ac \), called the discriminant, is positive.
• One real solution: If the discriminant is zero. This means the parabola touches the x-axis at exactly one point.
• Two complex solutions: If the discriminant is negative, resulting in imaginary numbers.

The quadratic formula lets you systematically work through these possibilities. It ensures that no matter the nature of the roots, you'll find the solutions.

Irrational Solutions

While solving a quadratic equation using the quadratic formula, you often encounter irrational numbers, represented by square roots. An "irrational solution" occurs when \( b^2 - 4ac \) leaves a non-perfect square under the square root. Simplification involves:

• Finding a simpler form of the square root where possible.
• Expressing your answer in simplest radical form. For example, \( \sqrt{8} \) simplifies to \( 2\sqrt{2} \).

In the given problem, the value under the square root turned to zero, leading to a simplified and rational solution. Always check if radical terms can be simplified further to ensure precise answers.

Mathematical Coefficients

Coefficients in a quadratic equation represent values that multiply the variables \( x \). They significantly influence the equation's solutions:

• \( a \): Determines the "width" and orientation of the parabola.
• \( b \): Impacts the position of the vertex along the x-axis.
• \( c \): Determines where the parabola crosses the y-axis.

In any quadratic equation, the values of \( a \), \( b \), and \( c \) need to be correctly identified and used in the quadratic formula to achieve the right answers. Misidentifying these coefficients can lead to incorrect solutions.