Problem 21 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations using the quadratic formula. If there are no real roots, state this as the answer. Exercises \(3-6\) are the same as Exercises \(13-16\) of Section 7.2. $$s^{2}=9+s(1-2 s)$$

Step-by-Step Solution

Answer

The solutions are \( s = \frac{1 + \sqrt{109}}{6} \) and \( s = \frac{1 - \sqrt{109}}{6} \).

Step 1: Rearrange Equation

First, rearrange the given equation in the standard form of a quadratic equation, which is \( ax^2 + bx + c = 0 \). The given equation is \( s^2 = 9 + s(1 - 2s) \). Expand and simplify the right side: \( s(1 - 2s) = s - 2s^2 \). Thus, \( s^2 = 9 + s - 2s^2 \). To rearrange, move all terms to one side: \( s^2 + 2s^2 - s - 9 = 0 \), simplifying to \( 3s^2 - s - 9 = 0 \).

Step 2: Identify Coefficients

Identify the coefficients from the quadratic equation \( 3s^2 - s - 9 = 0 \). Here, \( a = 3 \), \( b = -1 \), and \( c = -9 \). These coefficients will be used in the quadratic formula.

Step 3: Apply the Quadratic Formula

Use the quadratic formula \( s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the roots. Substitute the values: \( a = 3 \), \( b = -1 \), and \( c = -9 \) into the formula: \( s = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(3)(-9)}}{2(3)} \). Simplify under the square root: \( (-1)^2 = 1 \) and \(-4 \times 3 \times -9 = 108 \). Therefore, \( b^2 - 4ac = 1 + 108 = 109 \).

Step 4: Calculate the Roots

Now substitute into the formula: \( s = \frac{1 \pm \sqrt{109}}{6} \). Calculate the two solutions: \( s_1 = \frac{1 + \sqrt{109}}{6} \) and \( s_2 = \frac{1 - \sqrt{109}}{6} \).

Step 5: Conclude the Solution

Since the discriminant \( 109 \) is positive, there are two distinct real solutions. The solutions for the quadratic equation are \( s_1 = \frac{1 + \sqrt{109}}{6} \) and \( s_2 = \frac{1 - \sqrt{109}}{6} \).

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and appear frequently in various mathematical and real-world applications. A quadratic equation is a polynomial equation of degree two, typically expressed in the standard form: - \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero.
The graphs of quadratic functions are parabolas, which can open upwards or downwards depending on the sign of \( a \). Solving these equations is crucial, as they model many physical systems, ranging from the trajectory of a projectile to understanding economics in financial models. By manipulating and solving quadratic equations, students gain insight into how mathematical relationships are structured.
There are several methods to solve quadratic equations:

• Factoring, when simple integers are involved.
• Completing the square, which transforms the equation into a perfect square trinomial.
• The quadratic formula, which provides a direct solution by calculation.

Among these, the quadratic formula is versatile and works universally for any quadratic equation.

Real Roots

Real roots of a quadratic equation are the solutions where the graph of the equation intersects the x-axis. For the equation \( ax^2 + bx + c = 0 \), roots can be found graphically or algebraically. In simpler terms, the real roots are the x-values where the curve hits or touches the x-axis.
Not all quadratic equations have real roots. Whether or not a quadratic has real roots depends on the discriminant of the equation (this will be discussed in the next section).
- If the equation has two real roots, the parabola intersects the x-axis at two distinct points.- If there is one real root (also known as a repeated or double root), the parabola just touches the x-axis without crossing it.- If there are no real roots, the parabola does not touch the x-axis at all.This understanding of real roots is pivotal in both resolving quadratic equations and comprehending the graph's behavior.

Discriminant

The discriminant is a key element in determining the nature of the roots of a quadratic equation. For the equation \( ax^2 + bx + c = 0 \), the discriminant \( D \) is calculated using the formula:- \( D = b^2 - 4ac \).This discriminant reveals how the parabola related to the quadratic equation interacts with the x-axis. Its value indicates both the number and the nature of roots:

• If \( D > 0 \), the quadratic equation has two distinct real roots. This means the parabola crosses the x-axis at two points.
• If \( D = 0 \), there is exactly one real root, indicating the parabola is tangent to the x-axis at a vertex.
• If \( D < 0 \), there are no real roots; the graph does not intersect or touch the x-axis, indicating complex roots.

Understanding the discriminant is essential as it quickly offers a preview of the equation's solvability and the kind of solutions one can expect without solving the entire equation.