Problem 40 - A Graphical Approach to College Algebra

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$9 x^{2}-12 x=-8$$

Step-by-Step Solution

Answer

The equation has no real solutions, as its discriminant is negative.

Step 1: Rearrange the Equation

First, rearrange the equation to set it to zero. The given equation is:\[9x^2 - 12x = -8\]Add 8 to both sides of the equation to achieve:\[9x^2 - 12x + 8 = 0\]

Step 2: Identify Quadratic Coefficients

Recognize the quadratic form \(ax^2 + bx + c = 0\). Here, you have:- \(a = 9\)- \(b = -12\)- \(c = 8\)

Step 3: Apply the Quadratic Formula

The quadratic formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substitute the identified values of \(a\), \(b\), and \(c\):\[ x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \cdot 9 \cdot 8}}{2 \cdot 9} \]

Step 4: Calculate the Discriminant

Calculate the value inside the square root, known as the discriminant:\[b^2 - 4ac = (-12)^2 - 4 \cdot 9 \cdot 8 = 144 - 288 = -144\]Since the discriminant is negative, the equation has no real solutions.

Step 5: Provide Solution Explanation and Graphical Support

Since the discriminant is negative, it implies the equation has complex roots, which indicates that the solutions are not real numbers. Therefore, graphing will show that the parabola \(9x^2 - 12x + 8\) does not intersect the x-axis.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used for solving any quadratic equation of the form \(ax^2 + bx + c = 0\). It directly provides the solutions for the variable \(x\). Here's the formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Let’s break it down further:

• 'a', 'b', and 'c' are coefficients from the quadratic equation.
• The symbol \(\pm\) indicates the formula gives two possible solutions.
• The term under the square root, \(b^2 - 4ac\), is called the discriminant.

By substituting the values from the equation into this formula, you can find solutions whether they are real or complex. The presence of the square root means solutions can vary based on the discriminant's value.

Discriminant

The discriminant is a key component of the quadratic formula and is denoted by \(b^2 - 4ac\). It tells us crucial information about the nature of the solutions of a quadratic equation. Let's explore:

• When the discriminant is positive, \(b^2 - 4ac > 0\), the equation has two distinct real solutions.
• If it is zero, \(b^2 - 4ac = 0\), there is exactly one real solution (a repeated root).
• When negative, \(b^2 - 4ac < 0\), the solutions are complex or imaginary, and the graph does not intersect the x-axis.

For the exercise \(9x^2 - 12x + 8 = 0\), the discriminant calculated is -144, indicating complex solutions. Therefore, this equation doesn't have real number solutions.

Complex Solutions

Complex solutions arise when the discriminant of a quadratic equation is negative. When this is the case, the solutions will include imaginary numbers. Let’s look at this closer:

• The imaginary unit \(i\) is defined as \(\sqrt{-1}\).
• Complex solutions use this unit and can be represented as \(a + bi\), where \(a\) and \(b\) are real numbers.
• For our equation, the discriminant \(-144\) leads to solutions like \(\frac{12 \pm \sqrt{-144}}{18}\), translating to \(\frac{12 \pm 12i}{18}\).

This process converts a negative square root into a workable form employing 'i', enabling the expression and calculation of complex roots effectively. Such solutions indicate points on the graph beyond the real number space.