Problem 29 - Algebra 1: Concepts and Skills

Question

Find the value of \(b^{2}\)- 4ac for the equation. $$4 x^{2}+5 x+1=0$$

Step-by-Step Solution

Answer

The value of \(b^{2} - 4ac\) for the given quadratic equation is 9.

Step 1: Identify parameters

The given quadratic function is \(4x^{2}+5x+1=0\). In this equation, the coefficient of \(x^{2}\), \(x\), and the constant term are equivalent to \(a\), \(b\), \(c\) respectively in the general form of a quadratic equation \(ax^{2}+bx+c=0\). Therefore, from the given equation, \(a = 4\), \(b = 5\), and \(c = 1\).

Step 2: Apply the discriminant formula

The Discriminant (\(D\)) of a quadratic equation is found using the formula \(D = b^{2} - 4ac\). Put the values of \(a\), \(b\), \(c\) in the formula to find the value of \(D\).

Step 3: Calculation

By substituting the found values for \(a\), \(b\) and \(c\) into the discriminant formula, you get \(D = 5^{2} - 4*4*1 = 25 - 16\).

Step 4: Final Calculation

Calculating the above expression, we get \(D = 9\).

Key Concepts

Discriminant

The discriminant is a core concept when dealing with quadratic equations. It is found using the formula \( D = b^{2} - 4ac \). Determining the discriminant helps assess the nature of the roots for a quadratic equation.

• If \( D > 0 \), the equation has two distinct real roots.
• If \( D = 0 \), the equation has exactly one real root, meaning the roots are repeated.
• If \( D < 0 \), the equation does not have real roots but two complex roots.

When given the equation \( 4x^{2} + 5x + 1 = 0 \), we can use the values of \( a = 4 \), \( b = 5 \), and \( c = 1 \) to plug into the formula, leading to \( D = 5^{2} - 4 \times 4 \times 1 = 25 - 16 = 9 \). Since \( D = 9 \) is greater than zero, this means our equation has two distinct real roots.

Quadratic Formula

The quadratic formula is a reliable tool for finding the roots of any quadratic equation of the form \( ax^{2} + bx + c = 0 \). The formula is given by:\[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] This formula solves for \( x \) by using the coefficients \( a \), \( b \), and \( c \). The presence of the \( \pm \) symbol in the formula indicates that there can be two potential solutions.
Let's look at the connection to the discriminant again. The part under the square root, \( b^{2} - 4ac \), is the discriminant. As we learned earlier, the value of the discriminant determines the nature of the roots:

• If \( D > 0 \), the formula yields two real solutions because you are adding and subtracting a real number.
• If \( D = 0 \), adding and subtracting zero means both solutions are the same.
• If \( D < 0 \), the solutions are not real since the square root of a negative number involves imaginary numbers.

Thus, utilizing the quadratic formula, irrespective of the nature of your discriminant, will yield valid insight into the roots of the equation, helping solve any quadratic problem.

Coefficients

In the realm of quadratic equations, coefficients are the numerical factors preceding the terms in the expression. They play a pivotal role in shaping both the equation and its solution. For a standard quadratic equation \( ax^{2} + bx + c = 0 \), these coefficients are recognized as:

• \( a \): The coefficient of \( x^2 \), responsible for the curvature of the parabola graphically representing the quadratic. In our example \( a = 4 \).
• \( b \): The coefficient of \( x \), influences the symmetry axis of the parabola. Here, \( b = 5 \).
• \( c \): The constant term, impacts where the parabola intersects the y-axis in a graph. In the example given, \( c = 1 \).

Identifying and understanding these coefficients allows us not only to use the discriminant and quadratic formula effectively but also to predict the graph's shape and behavior. They determine the nature and number of graphically depicted solutions of the equation.