Problem 16 - Algebra 1: Concepts and Skills

Question

Find the discriminant of the quadratic equation. \(5 x^{2}+3 x=12\)

Step-by-Step Solution

Answer

The discriminant of the given quadratic equation is 249.

Step 1: Identify the Coefficients

The first step is to write the given equation in standard form. The given equation is \(5x^{2} + 3x = 12\). After rearranging, the equation becomes \(5x^{2} + 3x - 12 = 0\). For any quadratic equation in the form \(ax^{2} + bx + c = 0\), 'a' represents the coefficient of \(x^{2}\), 'b' represents the coefficient of 'x', and 'c' is the constant term. So in this case, a = 5, b = 3, and c = -12.

Step 2: Calculate the Discriminant

Next, the discriminant of a quadratic equation \(ax^{2} + bx + c = 0\) is given by the formula \(b^{2} - 4ac\). Substituting a = 5, b = 3, and c = -12 into the formula, we get \(D = b^{2} - 4ac= (3)^{2} - 4*5*(-12) = 249\).

Key Concepts

Quadratic Equations

Quadratic equations are the building blocks of algebra and they appear incessantly across various fields of mathematics and science. These equations fall into the category where the highest power of the unknown variable, usually denoted by 'x', is two. The general form of a quadratic equation is expressed as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) represent real numbers, with \( a \eq 0 \) since otherwise, it would not be a quadratic equation.

To illustrate, consider the equation \( 5x^2 + 3x = 12 \). Once it's converted into the standard form, as shown in the given solution, it clearly adheres to the shape of a quadratic equation, thereby allowing us to use specific methods developed for solving equations of this nature, such as factoring, completing the square, or employing the quadratic formula.

Standard Form Conversion

When we come across a quadratic equation, it's imperative to convert it into its standard form to utilize the tools available for their analysis and solution effectively. The standard form as mentioned is \( ax^2 + bx + c = 0 \), and this conversion makes it easier to identify the coefficients and the constant term, which are crucial for subsequent calculations, such as finding the discriminant or the roots of the equation.

In our example, the equation \( 5x^2 + 3x = 12 \) is not in standard form initially. By isolating all terms on one side, resulting in \( 5x^2 + 3x - 12 = 0 \), we can then easily deduce that \( a = 5 \), \( b = 3 \) and \( c = -12 \), setting us up for further steps such as calculating the discriminant, which can reveal the nature of the equation's roots.

Quadratic Formula

Beyond mere factorization or visualization, the quadratic formula is a powerful tool that gives us the roots of any quadratic equation directly. It is derived from the process of completing the square of the standard form of a quadratic equation and is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). The expression under the square root, \( b^2 - 4ac \), is known as the discriminant. It's denoted typically by 'D' or the Greek letter 'Delta'.

The discriminant is a critical value that determines the nature of the roots of the quadratic equation without actually solving for them. If \( D > 0 \), there are two distinct real roots. If \( D = 0 \), there is one real root (also called a repeated root). If \( D < 0 \), there are no real roots, and the solutions are complex numbers. By plugging in the coefficients from our example into the discriminant formula, we found \( D = 249 \) - this indicates our equation has two distinct real roots, thus offering a glimpse into the solution's nature even before we attempt to solve the quadratic equation entirely.