Problem 49 - Precalculus : Building Concepts and Connections

Question

Solve the quadratic equation by using the quadratic formula. Find only real solutions. $$-l^{2}+40 l=100$$

Step-by-Step Solution

Answer

The real solutions to the equation are \(l_1 = 20 - 10\sqrt(3)\) and \(l_2 = 20 + 10\sqrt(3)\)

Step 1: Identify the coefficients a, b, and c

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). In the equation \(-l^{2}+40 l=100\), we can rewrite this as \(-l^{2}+40 l-100=0\). Comparing this equation with the standard form, it can be inferred that \(a = -1\), \(b = 40\), and \(c = -100\).

Step 2: Calculate the discriminant

The discriminant of a quadratic equation is given by \(b^2 - 4ac\). Substituting the values of \(a = -1\), \(b = 40\), and \(c = -100\), we get \(\Delta = (40)^2 - 4*(-1)*(-100) = 1600 - 400 = 1200\). The discriminant is positive, so the equation has two distinct real roots.

Step 3: Find the roots of the equation with the quadratic formula

The quadratic formula is \(x = [-b ± sqrt(b^2 - 4ac)] / (2a)\). Substituting the values of a, b, c, and \(\Delta\) into the formula, the roots of the equation are \(l = [-(40) ± sqrt(1200)] / (2*-1)\). This simplifies to \(l_1 = 20 - 10\sqrt(3)\) and \(l_2 = 20 + 10\sqrt(3)\).

Key Concepts

Quadratic Formula

The quadratic formula is a critical tool for solving quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). This formula is a universal solution that provides the roots, or solutions, of any quadratic equation. It is expressed as:

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

The variables \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation. Here's how the quadratic formula works:

• "\(-b\)" reflects the symmetry of parabolas described by quadratic equations about the vertical axis.
• The square root component, "\(\sqrt{b^2 - 4ac}\)", determines the nature of the roots.
• The denominator, "\(2a\)", adjusts the scale of the solutions based on the width of the parabola.

The formula encompasses all possible outcomes for the quadratic function, whether it yields real or complex solutions. By substituting the appropriate values into the formula, you can derive the solutions to the equation.

Discriminant

The discriminant is a crucial component of the quadratic formula that appears under the square root component: \(\Delta = b^2 - 4ac\). It helps to determine the nature of the roots without solving the equation completely. Let's break it down:

• If \(\Delta > 0\), the equation has two distinct real solutions.
• If \(\Delta = 0\), there is exactly one real solution, also known as a repeated root.
• If \(\Delta < 0\), the equation has no real solutions, but rather two complex solutions.

Understanding the discriminant allows you to quickly assess not just the existence, but the nature of the solutions to a quadratic equation. In our exercise, the discriminant was calculated to be 1200, which is positive, indicating two distinct real roots.

Real Solutions

Real solutions are those values of \(x\) that satisfy the quadratic equation and are real numbers, rather than imaginary numbers. Real solutions occur when the discriminant is non-negative. In particular:

• When \(\Delta > 0\), you get two distinct real solutions which correspond to the points where the parabola intersects the x-axis.
• When \(\Delta = 0\), the parabola touches the x-axis at exactly one point, resulting in a single real solution or repeated root.

In our solved problem, because the discriminant is positive, the equation yields two real solutions: \(l_1 = 20 - 10\sqrt{3}\) and \(l_2 = 20 + 10\sqrt{3}\). These solutions show the specific values of \(l\) where the equation equals zero. This understanding is essential for determining the outcome of quadratic problems and evaluating potential answers.