Problem 28 - Algebra 1

Question

Find the value of \(b^{2}-4 a c\) for the equation. $$3 x^{2}-5 x-12=0$$

Step-by-Step Solution

Answer

The discriminant of the given quadratic equation is 169.

Step 1: Identify a, b and c

From the equation \(3x^{2}-5x-12=0\), we can identify that \(a=3\), \(b=-5\), and \(c=-12\).

Step 2: Substitute in the discriminant formula

Substitute the values of a, b, and c into the formula for the discriminant, \(b^{2}-4 a c\), giving us \((-5)^{2}-4*3*(-12)\).

Step 3: Calculate the value

Calculate the above expression to get the value of the discriminant. The squares of -5 is 25. Then multiply 4, 3 and -12 together, it yields 144. Adding both 25 and 144 gives a discriminant of 169.

Key Concepts

Quadratic Equation

A quadratic equation is a mathematical expression that includes a term with a variable raised to the second power. It takes the general form:

• \( ax^2 + bx + c = 0 \)

Here, \( x \) is the variable, and \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). The term with \( x^2 \) is what makes the equation quadratic.
Quadratics are a fundamental component of algebra due to their widespread applicability, including physics, engineering, and economics. Understanding how to analyze these equations allows us to find specific points known as roots, showing where the curve of the equation touches the horizontal axis.

Discriminant

The discriminant is a part of the quadratic equation's solutions. It is derived from the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

The discriminant is represented by \( b^2 - 4ac \). Its value indicates the nature of the roots of the quadratic equation.

Depending on its value, it tells us whether the equation has:

• Two distinct real roots (when the discriminant is positive)
• One real root (when the discriminant is zero)
• No real roots, but two complex roots (when the discriminant is negative)

Thus, the discriminant helps ascertain crucial information about the behavior of the quadratic equation without solving it completely.

Roots of Quadratic Equations

The roots of quadratic equations are the solutions to the equation set to zero. These roots are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \).
To find these roots, we typically use the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here, the expression under the square root, \( b^2 - 4ac \), is the discriminant.

The roots can display various characteristics:

• If the discriminant is positive, the equation has two different real roots.
• If the discriminant is zero, there is exactly one real (repeated) root.
• If the discriminant is negative, the equation has two complex conjugate roots.

These roots are the x-intercepts of the quadratic function's graph, providing essential insight into its shape and position on the coordinate plane.