Problem 31 - Intermediate Algebra

Question

Simplify and reduce each expression. $$ 3 x^{2}+19 x+20=0 $$

Step-by-Step Solution

Answer

The roots of the equation are \( x = \frac{-4}{3} \) and \( x = -5 \).

Step 1: Identify the Quadratic Equation

We are given the quadratic equation in the form \( ax^2 + bx + c = 0 \). Here, \( a = 3 \), \( b = 19 \), and \( c = 20 \).

Step 2: Apply the Quadratic Formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). We'll substitute the values of \( a \), \( b \), and \( c \) into this formula.

Step 3: Calculate the Discriminant

First, find the discriminant \( \Delta = b^2 - 4ac \). This helps determine the nature of the roots. Substitute \( b = 19 \), \( a = 3 \), and \( c = 20 \) to get \( \Delta = 19^2 - 4(3)(20) = 361 - 240 = 121 \).

Step 4: Determine the Roots Using the Discriminant

Since \( \Delta = 121 \) is a perfect square, the roots are real and rational. Use \( x = \frac{-19 \pm \sqrt{121}}{6} \), where \( \sqrt{121} = 11 \).

Step 5: Simplify for Each Root

Calculate each root. For the positive square root: \( x_1 = \frac{-19 + 11}{6} = \frac{-8}{6} = \frac{-4}{3} \). For the negative square root: \( x_2 = \frac{-19 - 11}{6} = \frac{-30}{6} = -5 \).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations of the form \( ax^2 + bx + c = 0 \). This formula is particularly useful when factoring is difficult or impossible. The formula is given as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This means that, by substituting the coefficients \( a \), \( b \), and \( c \) from the equation, we can effectively find the values of \( x \) that solve the equation. Let’s break it down:

• - **\( b \):** The coefficient of \( x \).
• - **\( a \):** The coefficient of \( x^2 \).
• - **\( c \):** The constant term.

If we correctly substitute each value into the formula, we can solve for \( x \), finding the roots of the equation. This technique provides a systematic approach to finding solutions even when equations are complex. Keep in mind that the next steps, like calculating the discriminant, hinge on this formula.

Discriminant

The discriminant is a key component of the quadratic formula, given by the expression \( b^2 - 4ac \). It plays a crucial role in determining the nature of the roots we might find from a quadratic equation. Here is why the discriminant is vital:

• - **It helps us determine the type of roots:**
  • If \( \Delta > 0 \), the roots are real and distinct.
  • If \( \Delta = 0 \), the roots are real and equal.
  • If \( \Delta < 0 \), the roots are complex and not real.
• - **Calculates the number of solutions:** As \( \Delta \) changes, so does the nature of the solutions.

In our specific equation, the discriminant \( \Delta = 121 \), which is greater than zero and a perfect square. This means that the equation has two distinct real and rational roots. Being able to calculate and interpret the discriminant is crucial for understanding the nature of your solutions.

Real and Rational Roots

When we talk about real and rational roots in the context of quadratic equations, we're referring to specific solutions where the values of \( x \) are both real numbers and rational numbers. Here's what that means:

• - **Real roots:** Solutions that exist on the number line. These can be plotted and seen visually.
• - **Rational roots:** Roots that can be expressed as fractions or whole numbers. Essentially, they can be written as \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q eq 0 \).

This specific quadratic equation, after evaluating its discriminant as a perfect square of 11, leads to roots \( x_1 = \frac{-4}{3} \) and \( x_2 = -5 \). Both are rational, meaning they are expressed via integer or fraction form. Understanding whether roots are real and rational can provide insights into further applications, such as graphing or real-world model predictions.