Problem 8 - College Algebra and Calculus: An Applied Approach

Question

Use the discriminant to determine the number of real solutions of the quadratic equation. \(\frac{1}{3} x^{2}-5 x+25=0\)

Step-by-Step Solution

Answer

By calculating and evaluating the discriminant of the given quadratic equation, one can determine the number of its real solutions. The results will be interpreted accordingly.

Step 1: Identify the coefficients

Identify the coefficients a, b, and c in the given quadratic equation \(\frac{1}{3} x^{2}-5 x+25=0\). Here, a = \(\frac{1}{3}\), b = -5, and c = 25.

Step 2: Calculate the Discriminant

Substitute the identified coefficients into the discriminant formula \(D = b^{2}-4ac\). So, D = \((-5)^{2}-4*\frac{1}{3}*25\)

Step 3: Evaluate the Discriminant

Evaluate the value of D. If the value of D is negative, the quadratic equation has no real solutions. If the value of D is equal to zero, the quadratic equation has one real solution, and if the value of D is positive, the quadratic equation has two real solutions.

Step 4: Interpret the result

The calculated D value will tell about the number of real solutions of the given quadratic equation.

Key Concepts

Quadratic Equation Solutions

Solving quadratic equations is a foundational skill in algebra. A quadratic equation is typically written in the form \( ax^2 + bx + c = 0 \) where \( a \) is not zero. To find the solutions of a quadratic equation, you can use factoring, completing the square, the quadratic formula, or graphing. Each of these methods may be more suitable in different scenarios, but they all hinge on the same basic principles.

The solutions of a quadratic equation are the values of \( x \) that make the equation true. These solutions are also called the 'roots' of the equation. Depending on the coefficients \( a \) , \( b \) , and \( c \) , a quadratic equation can have either two real solutions, one real solution, or no real solutions that are real numbers.

Calculating Discriminant

The discriminant is a vital part of the quadratic formula and provides crucial information about the nature of the solutions of a quadratic equation without actually solving it. The discriminant \( D \) is expressed as \( D = b^2 - 4ac \) where \( a \) , \( b \) , and \( c \) are the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \).

Calculating the discriminant is straightforward: substitute the coefficients from the quadratic into the discriminant formula and simplify. This single number will give you a wealth of information about the equation's solutions. A positive discriminant indicates two distinct real solutions, zero means exactly one real solution (also known as a repeated root), and a negative discriminant tells us there are no real solutions, implying that the solutions are complex numbers.

Real Solutions of Quadratic Equations

The real solutions of a quadratic equation correspond to the points where the equation's graph intersects with the \( x \) -axis. These intersections are the 'x-intercepts' or 'zeros' of the function.

When the discriminant \( D = b^2 - 4ac \) is positive, we have two different x-intercepts, indicating two distinct real solutions. If the discriminant is equal to zero, the graph touches the \( x \) -axis at a single point, which means there's exactly one real solution. It's also known as a double root since it’s one solution counted twice. Lastly, if the discriminant is negative, the quadratic's graph does not touch or cross the \( x \) -axis at all, signifying that there are no real solutions, and the solutions are instead complex numbers with imaginary parts. Understanding the discriminant is an invaluable tool for predicting the nature of a quadratic equation's solutions before even graphing it or solving it algebraically.