Problem 10 - Elementary Algebra

Question

Calculate the discriminant and use it to determine the number and type of solutions. Do not solve. $$ 12 x 2-x-12=0 $$

Step-by-Step Solution

Answer

The equation has two distinct real solutions.

Step 1: Identify the Coefficients

For the quadratic equation in the standard form \(ax^2 + bx + c = 0\), identify the coefficients for \(a\), \(b\), and \(c\). The given equation is \(12x^2 - x - 12 = 0\). Therefore, \(a = 12\), \(b = -1\), and \(c = -12\).

Step 2: Write the Discriminant Formula

The discriminant \(D\) of a quadratic equation is given by the formula \(D = b^2 - 4ac\). This formula will help us determine the number and types of solutions.

Step 3: Substitute and Calculate the Discriminant

Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula: \[D = (-1)^2 - 4 \cdot 12 \cdot (-12)\]. Calculate \((-1)^2 = 1\) and \(-4 \times 12 \times (-12) = 576\). Thus, \(D = 1 + 576 = 577\).

Step 4: Interpret the Discriminant

Since the discriminant \(D = 577\) is greater than zero, this indicates that the quadratic equation has two distinct real solutions.

Key Concepts

Understanding Quadratic Equations

A quadratic equation is a type of polynomial equation that is expressed in the standard form \( ax^2 + bx + c = 0 \). It is characterized by its highest exponent being 2, meaning the variable \( x \) is squared.
The solutions to a quadratic equation are the values of \( x \) that make the equation true, turning it ideally into \( 0 = 0 \). While there are various methods to solve quadratic equations, determining the type and number of solutions can often be done by calculating the discriminant, without needing to find the exact solutions.
This capability makes quadratic equations fundamental in mathematics, as they frequently appear in various practical and theoretical problems. Understanding their solutions can, therefore, provide insights into a wide range of scientific and engineering puzzles.

Real Solutions and the Discriminant

Solutions to quadratic equations can be categorized into different types based on the nature of their roots. One of the quickest ways to determine the type of solutions without actually solving the equation is by calculating the discriminant \( D \). The discriminant is derived from the coefficients of the quadratic equation and is given by the formula:
\( D = b^2 - 4ac \), where \( a \), \( b \), and \( c \) are the coefficients of \( x^2 \), \( x \), and the constant term, respectively.
The value of the discriminant tells us:

• If \( D > 0 \), the equation has two distinct real solutions.
• If \( D = 0 \), there is exactly one real solution or a repeated root.
• If \( D < 0 \), the solutions are non-real or complex numbers.

In our example, since \( D = 577 \), which is greater than zero, the quadratic equation has two distinct real solutions. This tells us that the graph of the equation will intersect the x-axis at two points.

Coefficients in Quadratic Equations

Every quadratic equation includes essential components known as coefficients, which influence the parabola's shape and position when the equation is graphed on a coordinate plane.
For the general form \( ax^2 + bx + c = 0 \):

• \( a \) is the coefficient of \( x^2 \) and determines the opening direction and width of the parabola. If \( a \) is positive, the parabola opens upwards, and if negative, it opens downwards.
• \( b \) is the coefficient of \( x \) and influences the parabola's tilt and direction.
• \( c \) is the constant term and affects the y-intercept, where the parabola touches the y-axis.

In the equation \( 12x^2 - x - 12 = 0 \), the coefficients are \( a = 12, b = -1, \) and \( c = -12 \). Knowing these coefficients allows us to calculate the discriminant and understand the properties of the solutions, including the number and nature of the roots.