Problem 85 - College Algebra

Question

Solve each equation in Exercises 73-98 by the method of your choice. \(3 x^{2}-12 x+12=0\)

Step-by-Step Solution

Answer

The solution to the equation \(3x^2 - 12x + 12 = 0\) is \(x = 2\).

Step 1: Identify a, b and c

The quadratic equation is \(3x^2 - 12x + 12 = 0\). So, \(a = 3\), \(b = -12\), and \(c = 12\).

Step 2: Plugging in the values into the quadratic formula

Plugging the values of \(a\), \(b\), and \(c\) into the quadratic formula \(-b \pm \sqrt{b^2 - 4ac} \over 2a\), we get \[12 \pm \sqrt{(-12)^2 - 4*3*12} \over 2*3.\] This simplifies to \[12 \pm \sqrt{144 - 144} \over 6\].

Step 3: Simplify the equation

This further simplifies to \[12 \pm \sqrt{0} \over 6\], which gives us \(x = 2\). The \( \pm \) in the quadratic formula usually gives us two solutions for \(x\), but in this case, as the discriminant (\(b^2 - 4ac\)) is 0, there is only one real solution for \(x\).

Key Concepts

Discriminant

The discriminant is a crucial component of the quadratic equation that helps us determine the nature of the solutions. When dealing with a quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant is found using the formula \(b^2 - 4ac\).

What the Discriminant Tells Us:

• If the discriminant is positive, there are two distinct real solutions.
• If it is zero, there is exactly one real solution.
• If the discriminant is negative, the solutions are not real numbers—they are complex or imaginary.

In our example, with \(a = 3\), \(b = -12\), and \(c = 12\), the discriminant is \((-12)^2 - 4 \times 3 \times 12 = 144 - 144 = 0\).

Hence, the equation has exactly one real solution. Understanding the discriminant helps in predicting whether completing the solution will provide us with real numbers or if we need to explore complex number solutions.

Quadratic Formula

The quadratic formula is a reliable method for finding the roots of any quadratic equation. It is expressed as:\[{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}.\]
Steps to Use the Quadratic Formula:

• Identify the coefficients \(a\), \(b\), and \(c\) from your quadratic equation.
• Substitute these values into the formula.
• Simplify under the square root and solve for \(x\).

In the problem \(3x^2 - 12x + 12 = 0\), we identify \(a = 3\), \(b = -12\), and \(c = 12\).
Substituting into the quadratic formula:\[x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \times 3 \times 12}}{2 \times 3} = \frac{12 \pm \sqrt{144 - 144}}{6}.\]
This simplifies to \(x = \frac{12 \pm 0}{6}\), yielding \(x = 2\). The quadratic formula simplified the solution process and confirmed the value found using the discriminant.

Real Solutions

Real solutions refer to the roots of the equation that are real numbers. In quadratics, these solutions determine where the graph of the equation touches or crosses the x-axis.

Types of Real Solutions:

• Two Distinct Real Solutions: Happens when the discriminant is positive.
• One Real Solution: Occurs when the discriminant is zero, resulting in a perfect square under the root which leads to one solution.

For the equation \(3x^2 - 12x + 12 = 0\), the discriminant turned out to be zero, leading us to one real solution \(x = 2\). This means that the graph of this quadratic equation merely touches the x-axis at one point, indicating a repeating root.

Understanding real solutions is vital, as they provide insight into the behavior of the parabola represented by the quadratic function.