Problem 28 - Algebra 1: Concepts and Skills

Question

Find the value of \(b^{2}\)- 4ac for the equation. $$x^{2}-3 x-4=0$$

Step-by-Step Solution

Answer

The value of the discriminant \(b^{2} - 4ac\) for the equation \(x^{2} - 3x - 4 = 0\) is 25.

Step 1: Identifying Coefficients

First, we identify the coefficients in the equation \(x^{2}-3x-4=0\). Here, \(a\) is the coefficient of \(x^{2}\), which is 1. Similarly, \(b\) is the coefficient of \(x\), which is -3, and \(c\) is the constant term, which is -4.

Step 2 : Calculate Discriminant

Next, we plug these coefficients into the formula for the discriminant, which is \(b^{2} - 4ac\). Substituting the values, we have \((-3)^{2}-4*1*(-4)\).

Step 3: Simplify the Expression

Simplifying \((-3)^{2}-4*1*(-4)\) we get \(9+16 = 25\).

Key Concepts

Identifying Coefficients

In a quadratic equation of the form \( ax^2 + bx + c = 0 \), identifying the coefficients is a fundamental first step in solving the equation. These coefficients are represented by \( a \), \( b \), and \( c \). Let's break down how to recognize these:

• **\( a \)**: The coefficient of \( x^2 \). In our example \( x^2 - 3x - 4 = 0 \), \( a = 1 \).
• **\( b \)**: The coefficient of \( x \). Here, \( b \) is The number accompanying \( x \), so \( b = -3 \).
• **\( c \)**: The constant term without any variable. For our equation, \( c = -4 \).

After identifying these coefficients, they can be utilized in other formulas and steps, such as calculating the discriminant or applying the quadratic formula.

Quadratic Formula

The quadratic formula is a splendid mathematical tool used to find the roots of a quadratic equation. The general representation of the quadratic formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula stems from the process of completing the square and can be employed to solve any quadratic equation. Key aspects to remember:

• **Discriminant \( b^2 - 4ac \)**: This part of the formula determines the nature of the roots. If the discriminant is positive, there will be two real and distinct roots. If it's zero, there will be one real root (or a repeated root). If it's negative, the roots will be complex or imaginary.
• The formula helps solve equations even when factoring or other simpler methods fail.

Always start with identifying \( a \), \( b \), and \( c \) before plugging in values into the formula. This preparation is crucial to avoid mistakes.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of \( x \) that make the equation equal zero. There are various methods including factoring, using the quadratic formula, and completing the square. Each method has its own advantages, but using the quadratic formula is a strong universal approach.The solution approach generally will include:

• **Using the quadratic formula**: As discussed, substituting \( a \), \( b \), and \( c \) in the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) after calculating the discriminant.


• **Checking results**: After finding the value(s) of \( x \), you can plug back these values into the original equation to verify if they meet the equation \( ax^2 + bx + c = 0 \).

This method's ultimate efficacy lies in its ability to solve any quadratic equation, ensuring you always find the roots, irrespective of their nature - real or complex.