Problem 38 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. \(31-44\) : Solve by using the quadratic formula. $$ \frac{5}{3} s^{2}+3 s+1=0 $$

Step-by-Step Solution

Answer

The solutions are \( s = \frac{-9 \pm \sqrt{21}}{10} \).

Step 1: Identify the coefficients

The quadratic equation given is \( \frac{5}{3}s^2 + 3s + 1 = 0 \). To use the quadratic formula, identify the coefficients: \( a = \frac{5}{3} \), \( b = 3 \), and \( c = 1 \).

Step 2: Write the quadratic formula

Recall the quadratic formula is used to find the roots of the equation \( ax^2 + bx + c = 0 \) and is given by:\[ s = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]

Step 3: Calculate the discriminant

The discriminant \( \Delta \) is \( b^2 - 4ac \). Substitute the coefficients into the formula:\[ \Delta = 3^2 - 4 \cdot \frac{5}{3} \cdot 1 = 9 - \frac{20}{3} = \frac{27}{3} - \frac{20}{3} = \frac{7}{3} \]

Step 4: Substitute into the quadratic formula

Substitute \( a \), \( b \), and \( \Delta \) back into the quadratic formula to solve for \( s \):\[ s = \frac{{-3 \pm \sqrt{\frac{7}{3}}}}{2 \cdot \frac{5}{3}} \]

Step 5: Simplify the expression

Simplify the fraction for the roots:\[ s = \frac{{-3 \pm \sqrt{\frac{7}{3}}}}{\frac{10}{3}} = \frac{{-3(3) \pm \sqrt{21}}}{10} = \frac{{-9 \pm \sqrt{21}}}{10} \]

Step 6: State the solutions

The solutions to the quadratic equation are:\( s = \frac{-9 + \sqrt{21}}{10} \quad \text{and} \quad s = \frac{-9 - \sqrt{21}}{10} \)

Key Concepts

Quadratic Equation

A quadratic equation is a special type of polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The equation is characterized by the \( x^2 \) term, which makes it a second-degree polynomial. Quadratic equations contain two solutions, known as roots, which can be found using various methods, with the quadratic formula being a prominent one. The purpose of solving a quadratic equation is to find the values of \( x \) that make the equation true, often referred to where the parabola, formed by the equation, intersects the x-axis. When solving, the value of \( a \) must not be zero, or else the equation becomes linear, lacking the quadratic component.

Coefficients

Coefficients are the numerical or constant multipliers of the variables in equations. In the context of the quadratic equation, they are the constants \( a \), \( b \), and \( c \). Each coefficient holds a particular role:

• \( a \): The coefficient of \( x^2 \), it affects the parabola's opening direction and width. The sign of \( a \) (positive or negative) determines if the parabola opens upwards or downwards.
• \( b \): The coefficient of \( x \), it impacts the parabola's axis of symmetry and influences the position of its vertex.
• \( c \): The constant term, it represents the y-intercept, or the point where the parabola crosses the y-axis.

Identifying these coefficients correctly is essential for applying the quadratic formula and obtaining the roots of the equation.

Discriminant Calculation

The discriminant of a quadratic equation is found in the expression \( b^2 - 4ac \), which is part of the quadratic formula under the square root. The value of the discriminant is key because it reveals the nature of the roots:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, the equation has exactly one real root, meaning both roots are the same.
• If the discriminant is negative, the roots are complex or imaginary, indicating no real solutions exist.

For the equation in question, the discriminant \( \Delta = \frac{7}{3} \) is positive, confirming the presence of two distinct real roots. This calculation informs us about the number and type of solutions in advance, guiding the subsequent solving steps.

Roots of Equation

The roots of a quadratic equation are the solutions found by substituting the coefficients into the quadratic formula:\[ s = \frac{{-b \pm \sqrt{b^2 - 4ac}}}{2a} \]This process involves substituting identified coefficients into the formula to compute the values of \( s \), the variable. The roots are represented as:

• \( s_1 = \frac{-9 + \sqrt{21}}{10} \)
• \( s_2 = \frac{-9 - \sqrt{21}}{10} \)

These solutions represent the precise points where the quadratic equation, or the corresponding parabola, intersects the x-axis, confirming the real number solutions for the equation given. Solving for the roots effectively concludes the process of solving the quadratic equation.