Problem 25 - Algebra 1

Question

Find the value of \(b^{2}-4 a c\) for the equation. $$x^{2}-11 x+30=0$$

Step-by-Step Solution

Answer

The value of \(b^2 - 4ac\) is 1.

Step 1: Identify the coefficients

The equation is \(x^{2}-11x+30=0\). Hence, \(a = 1\), \(b = -11\), \(c = 30\) from the general form of a quadratic equation \(ax^2 + bx + c = 0\).

Step 2: Apply the discriminant formula

Substitute the values of \(a\), \(b\), \(c\) in the formula \(b^2 - 4ac\), which gives us \((-11)^2 - 4(1)(30)\).

Step 3: Calculate the discriminant

Perform the calculation: \(121 - 120 = 1\).

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions of the second degree that can be written in the standard form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are known as the coefficients, and \(x\) represents the variable. These equations are fundamental in algebra and find applications across various fields such as physics, engineering, and economics.

The solutions to a quadratic equation, also known as roots or zeros, represent the values of \(x\) for which the equation is satisfied. These solutions can be real or complex numbers and can be determined through methods such as factoring, completing the square, or using the quadratic formula. An interesting aspect of quadratic equations is that they have a parabolic graph, with the vertex being the maximum or minimum point, depending on the sign of the \(a\) coefficient.

Discriminant Calculation

The discriminant of a quadratic equation is a valuable numerical expression that provides insights into the nature of the roots without actually solving the equation. It is represented by the symbol \(\Delta\) and calculated using the formula \(\Delta = b^2 - 4ac\), derived from the coefficients \(a\), \(b\), and \(c\) of the quadratic equation.

The discriminant can reveal the number and type of roots the quadratic equation has: if \(\Delta > 0\), the equation has two distinct real roots; if \(\Delta = 0\), there is one real root; and if \(\Delta < 0\), the roots are complex and conjugate. Understanding how to compute the discriminant is key to quickly assessing the properties of the quadratic equation without going through the complete solution process.

Algebraic Coefficients

Algebraic coefficients are the numerical or literal factors that multiply the variable terms within an algebraic expression or equation. In the context of a quadratic equation \(ax^2 + bx + c = 0\), \(a\) is the coefficient of the \(x^2\) term, \(b\) is the coefficient of the \(x\) term, and \(c\) is the constant term. These coefficients dictate the shape and location of the graph of the quadratic equation when plotted.

For example, the value of \(a\) determines whether the parabola opens upward or downward, while \(b\) and \(c\) influence the position of the vertex and the parabola along the x-axis and y-axis, respectively. Coefficients are essential in determining the discriminant and ultimately influence the nature and number of solutions to the equation.