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Discriminant of a Quadratic Equation

Do you want to know how many solutions a quadratic equation has without solving it? If yes, then you are in the right place! A quadratic equation is an equation that can be expressed in the form \(ax^2 + bx + c = 0\). In this blog post, we will discuss how to find the discriminant of a quadratic equation using the formula \(b^2 – 4ac\).

How to Find The Discriminant of a Quadratic Equation

The polynomial must first be written in standard form to calculate the discriminant. This involves organizing the polynomial so that its terms are arranged in descending powers of the variable.

Next, the coefficients are determined by comparing the variables to their corresponding exponents. Specifically, the coefficient of the quadratic term \((x^2)\) is denoted by ‘a’, the linear term \((x)\) coefficient is denoted by ‘b’, and the constant term is denoted by ‘c’.

Once the coefficients are identified, they can be substituted into the discriminant formula given by \(Δ = b^2 – 4ac\). 

Solving this formula yields the discriminant, a numerical quantity that reveals the nature of the roots of the quadratic equation.

Cases of Discriminant 

The value of the discriminant can assume one of three possible signs: positive, zero, or negative. The sign of the discriminant is a key factor in determining the number of solutions to the corresponding quadratic equation.

Positive Discriminant

For a quadratic equation with a positive discriminant, two real roots correspond to the points where the equation graph intersects the x-axis. Thus, a positive value of the discriminant guarantees that the quadratic has two distinct x-intercepts, which implies that the graph must cross the x-axis twice.

Furthermore, the sign of the coefficient of the quadratic term affects the shape of the graph. 

• If the coefficient a is positive, the graph is concave upward and has a minimum point below the x-axis
• If the coefficient a is negative, the graph is concave downward and has a maximum point above the x-axis

Zero Discriminant

For a quadratic equation with a zero discriminant, one real root corresponds to the point where the equation graph touches the x-axis. Thus, a discriminant of zero guarantees that the quadratic has only one x-intercept, which implies that the graph must touch the x-axis at its minimum or maximum point.

Moreover, when the discriminant is zero, the graph of the quadratic equation cannot pass through the x-axis, but instead, it touches it at the unique root. 

• If the coefficient a of the quadratic term is positive, the graph is concave upward, and the minimum point of the graph touches the x-axis. 
• If the coefficient a is negative, the graph is concave downward, and the maximum point of the graph touches the x-axis.

Negative Discriminant

When a quadratic equation has a negative discriminant, indicated by a value of \(b^2 – 4ac\) that is less than zero, the quadratic formula involves taking the square root of a negative number. Since the square root of a negative number is not real, there are no real solutions to the quadratic equation. Instead, the two solutions are complex and cannot be represented on a graph using real coordinates.

Geometrically, a negative discriminant means that the quadratic equation graph does not intersect the x-axis at any point. 

• If the coefficient a of the quadratic term is positive, the graph is concave upward, and the entire graph lies above the x-axis. This means that all outputs of the graph are positive. 
• If the coefficient a is negative, the graph is concave downward, and the entire graph lies below the x-axis. Consequently, all outputs of the graph are negative.

Example

For the equation \(x^2 + 5x + 6 = 0\), 

we have \(a = 1\), \(b = 5\), and \(c = 6\).

So, the discriminant D is:

\(D = b^2 – 4ac\)

\(= 5^2 – 4(1)(6)\)

\(= 25 – 24\)

\(= 1\)

Since the discriminant D is 1, which is positive, the quadratic equation has two distinct real roots.