Problem 4 - 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. $$x^{2}-x-13=0$$

Step-by-Step Solution

Answer

The approximate roots of the quadratic equation \(x^2 - x - 13 = 0\) are \(x_1 \approx 4.140\) and \(x_2 \approx -3.140\).

Step 1: Apply the Quadratic Formula

To find the roots of the quadratic equation, apply the quadratic formula, which states that for any quadratic equation in the form of \(ax^2 + bx + c = 0\), the solution can be found by using the formula: \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]. For our equation \(x^2 - x - 13 = 0\), where \(a = 1\), \(b = -1\), and \(c = -13\), plugging these values into the formula gives us: \[x = \frac{{-(-1) \pm \sqrt{{(-1)^2 - 4(1)(-13)}}}}{2(1)}\].

Step 2: Calculate the Discriminant

Calculate the discriminant \(\Delta\) using the part under the square root of the quadratic formula: \(\Delta = b^2 - 4ac\). Substituting our values in, we get: \[\Delta = (-1)^2 - 4(1)(-13) = 1 + 52 = 53\].

Step 3: Evaluate the Quadratic Formula

Evaluate the two possible solutions using the quadratic formula. With the discriminant \(\Delta = 53\), we have: \[x = \frac{{1 \pm \sqrt{53}}}{{2}}\]. This gives us two roots, \[x_1 = \frac{{1 + \sqrt{53}}}{2}\] and \[x_2 = \frac{{1 - \sqrt{53}}}{2}\].

Step 4: Approximate the Roots

Approximate the roots to three significant digits. Using a calculator, we can approximate: \[x_1 \approx \frac{1 + 7.280}{2} \approx 4.140\] and \[x_2 \approx \frac{1 - 7.280}{2} \approx -3.140\]. These are the approximate roots of the quadratic equation \(x^2 - x - 13 = 0\).

Key Concepts

Finding Roots of a Quadratic Equation

The process of finding roots of a quadratic equation is essential in solving quadratic equations. Roots are the solutions to the equation where it equals zero and may be real or complex numbers.

The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \) where \( a \) is the coefficient of \( x^2 \) (must not be zero), \( b \) is the coefficient of \( x \) and \( c \) is the constant term. There are several methods to find the roots, such as factoring, using the quadratic formula, completing the square, or graphing. However, the most universal method is the quadratic formula as it can be applied when other methods fail or are inefficient.

In the case of our example \( x^2 - x - 13 = 0 \), the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is the most feasible approach. Following the formula, we solve for \( x \) to find the roots, resulting in two potential solutions based on the use of \( \pm \) in the equation.

Discriminant Calculation

The discriminant is a key component in the quadratic formula, represented by the symbol \( \Delta \) and found within the square root of the formula: \( \Delta = b^2 - 4ac \).

The value of the discriminant gives us information about the nature of the roots:

• If \( \Delta > 0 \) there are two distinct real roots.
• If \( \Delta = 0 \) there is exactly one real root (also known as a repeated or double root).
• If \( \Delta < 0 \) there are no real roots, but two complex roots.

For our given equation \( x^2 - x - 13 = 0 \), the discriminant calculation step reveals \( \Delta = 53 \) after plugging in the values for \( a \) (-1), \( b \) (1), and \( c \) (-13). Since \( \Delta \) is greater than zero, it indicates that two distinct real roots exist for this quadratic equation.

Quadratic Equation Solutions

Upon determining the discriminant, the next step is to solve for the quadratic equation solutions, which yields the roots. Depending on the value of \( \Delta \), the expression \( \pm \sqrt{\Delta} \) in the quadratic formula will affect the resultant roots.

When \( \Delta \) is positive, like our example with \( \Delta = 53 \) for the equation \( x^2 - x - 13 = 0 \), we get two real solutions. These solutions or roots are found by evaluating \( x_1 = \frac{1 + \sqrt{53}}{2} \) and \( x_2 = \frac{1 - \sqrt{53}}{2} \), resulting from the 'plus-or-minus' portion of the quadratic formula.

Approximating these to three significant digits is often needed for practicality and simplicity in many applications. Hence, we calculate \( x_1 \) to be approximately 4.140 and \( x_2 \) to be approximately -3.140, which are the approximate solutions to the equation.