Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. \(31-44\) : Solve by using the quadratic formula. $$ 5 x^{2}+13 x=6 $$

Step-by-Step Solution

Answer

The solutions are \( x = 0.4 \) and \( x = -3 \).

Step 1: Write the Equation in Standard Form

First, ensure the quadratic equation is in the standard form, \( ax^2 + bx + c = 0 \). The given equation is \( 5x^2 + 13x = 6 \). To get the standard form, subtract \( 6 \) from both sides, resulting in \( 5x^2 + 13x - 6 = 0 \).

Step 2: Identify Coefficients

From the equation \( 5x^2 + 13x - 6 = 0 \), identify the coefficients: \( a = 5 \), \( b = 13 \), and \( c = -6 \).

Step 3: Apply the Quadratic Formula

The quadratic formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute \( a = 5 \), \( b = 13 \), and \( c = -6 \) into the formula: \( x = \frac{-13 \pm \sqrt{13^2 - 4 \cdot 5 \cdot (-6)}}{2 \cdot 5} \).

Step 4: Calculate the Discriminant

Compute the discriminant \( b^2 - 4ac \): \( 13^2 - 4 \cdot 5 \cdot (-6) = 169 + 120 = 289 \).

Step 5: Solve for x

Substitute the discriminant back into the quadratic formula: \( x = \frac{-13 \pm \sqrt{289}}{10} \). Since \( \sqrt{289} = 17 \), the equation becomes \( x = \frac{-13 \pm 17}{10} \).

Step 6: Find the Roots

Calculate the roots by considering both \( + \) and \( - \):1. For \( x = \frac{-13 + 17}{10} \), \( x = \frac{4}{10} = 0.4 \).2. For \( x = \frac{-13 - 17}{10} \), \( x = \frac{-30}{10} = -3 \).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that is characterized by having a degree of two, meaning the highest power of the variable is squared. The general representation of a quadratic equation is:

• \( ax^2 + bx + c = 0 \)

where \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). This form is quintessential in mathematics as it allows for the comprehensive analysis and solution of many kinds of problems. Quadratic equations often arise in diverse fields such as physics, engineering, and economics.
Solving quadratic equations involves finding the values of \( x \) that satisfy the equation. These values are known as the roots of the quadratic equation. Methods such as factoring, completing the square, and the quadratic formula can be utilized to solve quadratic equations. Among these, the quadratic formula is a universal method that can solve any quadratic equation by providing an algebraic solution, regardless of its complexity.

Discriminant Calculation

The discriminant is a crucial component of a quadratic equation. It is represented by the expression \( b^2 - 4ac \), which is found under the square root in the quadratic formula. The discriminant tells us about the nature of the roots of the quadratic equation:

• If \( b^2 - 4ac > 0 \), the equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \), the equation has exactly one real root, also known as a repeated or double root.
• If \( b^2 - 4ac < 0 \), the equation has no real roots but two complex (conjugate) roots.

In our example, the discriminant was calculated as \( 289 \), a positive number, indicating the quadratic equation has two distinct real roots. Understanding the discriminant is essential as it provides insight into the possible solutions of the equation without explicitly solving it.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is a tidy way to express any quadratic equation. It is given by:

• \( ax^2 + bx + c = 0 \)

This form is integral for solving the equation via the quadratic formula, as it clearly presents the coefficients \( a \), \( b \), and \( c \) needed for various calculations. Moving the terms to one side and setting the equation to zero is the defining characteristic of the standard form.
In our example, the equation \( 5x^2 + 13x = 6 \) was initially not in standard form. By rearranging it to \( 5x^2 + 13x - 6 = 0 \), we achieved the standard format. This transformation is crucial as it allows easy identification and substitution of the coefficients into the quadratic formula, facilitating the calculation of the roots.