Problem 78 - College Algebra

Question

In Exercises \(75-82\), compute the discriminant. Then determine the number and type of solutions for the given equation. $$2 x^{2}+11 x-6=0$$

Step-by-Step Solution

Answer

The discriminant for the given quadratic equation is 169 which implies the equation has two distinct real roots.

Step 1: Identify A, B, and C Parameters

In the provided quadratic equation, coefficients are \(a=2, b=11, c=-6\). Using these coefficients the discriminant can be computed using the formula \(D= b^2 - 4ac\).

Step 2: Calculate the Discriminant

Plugging the values of a, b, and c into the discriminant formula: \[D= (11)^2 - 4(2)(-6) = 121 + 48 = 169\]

Step 3: Determine the Number and Type of Solutions

Since the discriminant \(D = 169\) which is greater than 0, the given quadratic equation has two distinct real roots.

Key Concepts

Understanding Quadratic Equations

A quadratic equation is a fundamental concept in algebra, appearing in many mathematical problems across different fields. It's usually expressed in the standard form:

• Standard Form: \( ax^2 + bx + c = 0 \)

Here, \(a\), \(b\), and \(c\) are coefficients where \(a eq 0\) because if \(a = 0\), the equation becomes linear rather than quadratic.
The "quadratic" in the equation refers to the term with the exponent 2, which is what makes this type of equation unique compared to linear equations.
Solving quadratic equations typically involves finding the value(s) of \(x\) that satisfy the equation. The methods include factorization, completing the square, and using the quadratic formula. Each method will yield the roots or solutions of the equation.

Exploring Real Roots

The solutions to a quadratic equation are also known as its roots. When dealing with real numbers, these roots can either be real or complex. To determine the nature of the roots, we use the discriminant.
The discriminant is calculated using the formula:

• \( D = b^2 - 4ac \)

The value of the discriminant tells us:

• If \(D > 0\): The equation has two distinct real roots.
• If \(D = 0\): The equation has exactly one real root (a repeated or double root).
• If \(D < 0\): The equation results in two complex (non-real) roots.

Real roots are the solutions that you can graph on a number line, making them crucial for many practical applications involving real-world data.

Solutions in Algebra

In algebra, finding the solutions of an equation is often the primary goal. For quadratic equations, solutions tell us the values of the variable \(x\) that make the equation true. The famous quadratic formula can solve any quadratic equation:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

• This formula uses the discriminant, \(b^2 - 4ac\), to determine the number and type of roots.

The \(\pm\) sign indicates that there are typically two solutions, reflecting the possible reality of two roots.
Understanding these solutions helps in the analysis of parabolic relationships in physics, engineering, and statistics where quadratic equations often model certain situations, like the trajectory of a projectile or profit maximization in a business context.