Problem 34 - College Algebra

Question

Determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve. $$ 3 x^{2}+5 x-8=0 $$

Step-by-Step Solution

Answer

Two distinct real rational solutions.

Step 1: Identify the coefficients

The given quadratic equation is \(3x^2 + 5x - 8 = 0\). Identify the coefficients: \(a = 3\), \(b = 5\), and \(c = -8\).

Step 2: Write the discriminant formula

The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(\Delta = b^2 - 4ac\).

Step 3: Substitute the values into the discriminant formula

Substitute \(a = 3\), \(b = 5\), and \(c = -8\) into the formula: \(\Delta = 5^2 - 4 \times 3 \times (-8)\).

Step 4: Calculate the discriminant

Calculate the discriminant: \(\Delta = 25 + 96 = 121\).

Step 5: Analyze the discriminant

Since the discriminant \(\Delta = 121\) is positive and a perfect square, the quadratic equation has two distinct real rational solutions.

Key Concepts

Quadratic Equations

A quadratic equation is an expression that equates to zero and is of the form \(ax^2 + bx + c = 0\). It is a polynomial equation of degree 2, meaning that the highest power of the variable \(x\) is 2. Quadratic equations are fundamental in algebra and appear in various mathematical problems and real-life applications.

When analyzing quadratic equations, the first step is to identify the coefficients \(a\), \(b\), and \(c\):

• \(a\) - the coefficient of \(x^2\)
• \(b\) - the coefficient of \(x\)
• \(c\) - the constant term

For instance, in the equation \(3x^2 + 5x - 8 = 0\), we have \(a = 3\), \(b = 5\), and \(c = -8\). Understanding these coefficients is crucial for solving the equation or determining its properties, such as its solutions.

Nature of Solutions

The nature of solutions of a quadratic equation is determined by its discriminant, \(\Delta\). The discriminant is calculated using the formula \(\Delta = b^2 - 4ac\). This value helps us understand how many solutions exist and the type of solutions they are.

Here's what the discriminant reveals:

• If \(\Delta > 0\), there are two distinct real solutions. If it's a perfect square, the solutions are rational.
• If \(\Delta = 0\), there's exactly one real solution, which is also rational.
• If \(\Delta < 0\), there are no real solutions, only two complex solutions.

For the quadratic equation \(3x^2 + 5x - 8 = 0\), the discriminant is 121. Since 121 is greater than 0 and a perfect square, it indicates two distinct real rational solutions.

Rational Solutions

Rational solutions are solutions of an equation that can be expressed as a fraction of two integers. When dealing with quadratic equations, determining the rationality of solutions is crucial for understanding the nature of the equation's roots.

A quadratic equation has rational solutions when its discriminant is a perfect square. This is because the roots can be expressed using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). If \(\sqrt{b^2 - 4ac}\) is an integer, then the entire solution simplifies to a rational number.

In the example \(3x^2 + 5x - 8 = 0\), the discriminant is 121, which is \(11^2\), a perfect square. As a result, the equation has two rational solutions, making it easier to express these roots in a simple fractional form. Rational solutions are especially useful in contexts where precise, finite decimal representation is needed.