Question

Find the discriminant of the quadratic equation and describe the number and type of solutions of the equation. \(-18 p=p^2+81\)

Short answer

The discriminant of the quadratic equation is 0, indicating that the equation has one real, repeated root.

Step-by-step solution

Identify the Coefficients a, b, and c

From the given quadratic equation -18p = p^2 + 81, we can identify the coefficients as follows: a = 1, b = 18, and c = 81.

Calculate the Discriminant

Use the discriminant formula \(\Delta = b^2 - 4ac\) to compute the discriminant. By inserting the values of a, b, c into the formula, we get: \(\Delta = (18)^2 - 4*(1)*(81) = 324 - 324 = 0\).

Determine the Type and Number of Solutions

Based on the value of the discriminant, we can determine the number and type of solutions of the quadratic equation. Given that \(\Delta = 0\), it follows that the equation has one real root (the root is repeated).

Key concepts

Quadratic Formula

The quadratic formula provides a straightforward method for solving any quadratic equation of the form ax^2 +bx + c=0. The formula is expressed as:
\[\begin{equation}x = \frac{{-b \pm \sqrt{b^2 - 4ac}}}{2a}\end{equation}\]
In this formula, 'x' represents the roots of the quadratic equation, 'a', 'b', and 'c' are the coefficients that correspond to the terms of the quadratic equation (the coefficient 'a' cannot be zero), and the expression under the square root, \(b^2 - 4ac\), is known as the discriminant.

To apply the quadratic formula effectively, first identify the coefficients from the given equation. For the equation \(-18 p = p^2 + 81\), reorganize it into standard form to get \(p^2 + 18p + 81 = 0\), where \(a = 1\), \(b = 18\), and \(c = 81\). Then, insert these coefficients into the quadratic formula to find the roots. In this case, the discriminant calculated is zero, which indicates a unique scenario we'll explore in the next section.

• The quadratic formula is a universal solver for quadratic equations.
• It's imperative to first ensure the equation is in standard form before identifying coefficients.
• The discriminant plays a crucial role in the quadratic formula, influencing the types and nature of roots.

Types of Solutions

Quadratic equations can yield different types of solutions, and the discriminant \(\Delta\) is key to determining the nature of these solutions.

• \(\Delta > 0\): If the discriminant is positive, there are two distinct real roots.
• \(\Delta = 0\): A zero discriminant indicates that there is one real root, which is also known as a repeated or double root.
• \(\Delta < 0\): A negative discriminant means there are no real roots; instead, there are two complex roots, which are conjugates of each other.

In our original exercise, the discriminant was found to be zero. This directly tells us that the quadratic equation \(p^2 + 18p + 81\) will have exactly one real root. Since there is only one solution, this is often referred to as a 'perfect square trinomial', resulting from the square of a binomial. The root is repeated, meaning the parabola touches the x-axis at just one point. Understanding the discriminant provides valuable insight into the graph of the equation and the possible solutions without necessarily computing the roots.

Quadratic Equation Roots

The roots of a quadratic equation are the values of 'x' that satisfy the equation \(ax^2 + bx + c = 0\).

For our equation \(-18p = p^2 + 81\), we've determined that the discriminant is zero. In such cases, we only have one root, which can be easily found by completing the square or simply applying the quadratic formula with the discriminant as zero.

Finding the Single Root

Since the discriminant is zero, the quadratic formula simplifies to \(x = -\frac{b}{2a}\). By plugging in our coefficients \(a = 1\) and \(b = 18\), we can quickly find this root. The solution signifies the point where the vertex of the parabola, represented by the quadratic equation, touches the x-axis.

Understanding the concept of roots is fundamental in algebra and has practical implications in fields such as physics, engineering, and economics, where quadratic relationships often arise. A single root implies a unique solution to the problem at hand, which in some contexts, could denote equilibrium or the point of optimization.

• The roots are the solutions to the quadratic equation.
• A zero discriminant leads to a single root, indicating a parabola with a vertex touching the x-axis.
• Quadratic roots are not just abstract concepts but have practical applications across various disciplines.