Problem 5 - Technical Mathematics with Calculus

Question

Find the roots of each quadratic by any of the methods shown in this section. Keep three significant digits. For some, use more than one method and compare results. Explicit Functions. $$3 x^{2}+12 x-35=0$$

Step-by-Step Solution

Answer

The roots are approximately \( x_1 \approx 1.958 \) and \( x_2 \approx -5.958 \).

Step 1: Apply the Quadratic Formula

Use the quadratic formula which is given by \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \) to find the roots. In the given equation, \(3 x^{2}+12 x-35=0\), the coefficients are \( a = 3 \), \( b = 12 \), and \( c = -35 \).

Step 2: Calculate the Discriminant

Calculate the discriminant using \( b^2 - 4ac \). This is \( 12^2 - 4 \cdot 3 \cdot (-35) = 144 + 420 = 564 \).

Step 3: Evaluate the Roots

With \( a = 3 \), \( b = 12 \), and the discriminant \( D = 564 \) calculated, find the two roots using the formula: \( x = \frac{{-b \pm \sqrt{D}}}{{2a}} \). These roots are \( x = \frac{{-12 \pm \sqrt{564}}}{{6}} \).

Step 4: Simplify the Roots

Simplify under the square root, \( \sqrt{564} \approx 23.748 \). The roots are then \( x \approx \frac{{-12 \pm 23.748}}{{6}} \).

Step 5: Find the Individual Roots

The two roots are found by solving for both the \( + \) and \( - \) in the formula. This gives \( x_1 \approx \frac{{-12 + 23.748}}{{6}} \approx 1.958 \) and \( x_2 \approx \frac{{-12 - 23.748}}{{6}} \approx -5.958 \).

Key Concepts

Solving Quadratic Equations

Quadratic equations are ubiquitous in mathematics, representing curvatures and parabolic trends in various fields. Solving them entails finding values of the variable, usually denoted as 'x', that make the equation true. These solutions are referred to as the 'roots' or 'zeroes' of the equation.

For a standard quadratic equation in the form of \(ax^2 + bx + c = 0\), there are several methods for finding the roots, including factoring, completing the square, graphing, and using the quadratic formula. The latter is a reliable tool that can solve any quadratic equation, including those that are not easily factorable. The quadratic formula, \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\), offers a direct pathway to the roots, leveraging the coefficients of the equation for computation.

Let's revisit the exercise example \(3 x^{2}+12 x-35=0\). By applying the quadratic formula, we identify the coefficients, calculate the discriminant, and use these to compute the roots. The step-by-step process showcased exemplifies the formula's effectiveness in systematically arriving at the solution, ensuring a methodical approach to understanding quadratic equations.

Discriminant of a Quadratic Equation

The discriminant is a powerful component within the quadratic formula, symbolized by 'D', and is derived from the standard form of a quadratic equation \(ax^2 + bx + c = 0\). It is given by the expression \(D = b^2 - 4ac\), and plays a pivotal role in determining the nature and number of the roots of the equation.

The discriminant reveals whether the roots are real or complex, and whether they are distinct or repeated. If \(D > 0\), the equation has two distinct real roots. If \(D = 0\), there is exactly one real root, also known as a repeated or double root. Conversely, if \(D < 0\), the roots are complex and conjugate pairs. For the example \(3x^2 + 12x - 35 = 0\), we compute the discriminant as \(D = 12^2 - 4\cdot3\cdot(-35) = 564\), indicating two distinct real roots. Knowing the discriminant gives us a clear forecast of the solutions even before we calculate the exact values of the roots.

Roots of a Quadratic Equation

The roots of a quadratic equation are the values at which the equation intercepts the x-axis on a graph, also understood as the solutions to the equation. After computing the discriminant, the quadratic formula leads us to the roots themselves.

In our exercise solution, by applying the plus-minus sign in \(x = \frac{{-b \pm \sqrt{D}}}{{2a}}\), we obtain two roots, reflecting the parabola's intersection with the x-axis at two distinct points. This computation provides us with an explicit understanding of the quadratic function's behavior and its graphical representation. In the exercise \(3 x^{2}+12 x-35=0\), we calculated the roots to be approximately 1.958 and -5.958, which are the x-coordinates where the parabola \(y = 3 x^{2}+12 x-35\) crosses the x-axis.

Explicit Functions

An explicit function explicitly defines the output variable for each input. In the context of quadratic equations, the explicit function is typically presented in the form \(y = ax^2 + bx + c\), where 'a', 'b', and 'c' are known coefficients and 'x' is the variable or input.

In the case of \(3 x^{2}+12 x-35=0\), the explicit function form would be \(y = 3x^2 + 12x - 35\), highlighting that for every value of 'x', there is a corresponding value of 'y' determined by the quadratic expression. Explicit functions are crucial for graphing as they provide a clear recipe for plotting the curve. When converted to the standard quadratic equation to find roots, we set the output 'y' to zero, as shown in the exercise, indicating the points where the graph crosses the x-axis. Understanding the concept of explicit functions aids in visualizing the solutions and offers a tangible way to interact with the parabolic shapes they represent.