Problem 6 - College Algebra and Calculus: An Applied Approach

Question

Use the discriminant to determine the number of real solutions of the quadratic equation. \(3-6 x=-3 x^{2}\)

Step-by-Step Solution

Answer

The quadratic equation has exactly one real solution.

Step 1: Put the equation in standard form

Rearrange the given equation into the standard quadratic form, \(ax^{2} + bx + c = 0\). The given equation is \(3-6x=-3x^{2}\). So, after rearranging the equation, it becomes \(3x^{2} - 6x + 3 = 0\). Now, \(a = 3\), \(b = -6\) and \(c = 3\).

Step 2: Compute the discriminant

Use the coefficients in the standard form to evaluate the discriminant using the formula \(b^{2}-4ac\). Therefore, discriminant = \((-6)^{2}-4*3*3 = 36 - 36 = 0\).

Step 3: Determine the number of solutions

Use the discriminant to determine the number of real solutions. Since the discriminant equals 0, this means the equation has exactly one real solution.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of degree two, which means its highest exponent is two.
It generally takes the form:
\[ ax^2 + bx + c = 0 \]where:

• \(a\), \(b\), and \(c\) are coefficients
• \(a eq 0\) (if \(a = 0\), the equation becomes linear, not quadratic)
• \(x\) is the variable or unknown which we are trying to solve for

A quadratic equation can have two solutions because the graph of a quadratic is a parabola, which can intersect the x-axis at most two times. When solving, these solutions are often referred to as "roots" or "zeroes" of the equation, and they can be real or complex numbers depending on the discriminant value.

Real Solutions

When considering real solutions of a quadratic equation, the discriminant plays a crucial role in determining how many (and what type of) solutions exist.
The discriminant, derived from the coefficients of the quadratic equation in standard form, is:
\[ b^2 - 4ac \]This value decides the nature of the roots based on the following conditions:

• If the discriminant is positive (\(> 0\)), there are two distinct real solutions.
• If the discriminant is zero (\(= 0\)), there is exactly one real solution, also known as a double root.
• If the discriminant is negative (\(< 0\)), there are no real solutions; instead, there are two complex solutions.

In the given problem, since the discriminant was found to be zero, it indicates that the quadratic equation has exactly one real solution.

Standard Form

The standard form of a quadratic equation is a foundational concept for solving these types of equations.
It allows us to easily apply formulas and methods to find the solutions. The standard form is:
\[ ax^2 + bx + c = 0 \]
To convert any quadratic equation to this form, you might need to rearrange terms or simplify the expression. This makes it straightforward to identify the values of \(a\), \(b\), and \(c\), which are integral for calculating the discriminant.

For example, in the provided exercise, the original equation was rearranged from:
\(3 - 6x = -3x^2\)
to:
\(3x^2 - 6x + 3 = 0\)
This placed the equation into the standard form, allowing us to correctly identify the coefficients and proceed with evaluating the discriminant for solutions.