Problem 78 - Algebra and Trigonometry

Question

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 of the given quadratic equation is 169. Hence, the quadratic equation has two distinct real solutions.

Step 1: Identify coefficients of the quadratic equation

From the quadratic equation \(2x^2 + 11x - 6 = 0\), the coefficients can be identified as follows: \(a=2\), \(b=11\), and \(c=-6\).

Step 2: Calculate the Discriminant

The discriminant can be calculated using the formula \(D = b^2 - 4ac\). Substituting our coefficients into the formula, we get: \(D = (11)^2 - 4*(2)*(-6)= 121+48 = 169\).

Step 3: Determine the number and type of solutions

The value of the discriminant tells us about the number and type of solutions to the quadratic equation. If the discriminant is greater than zero, there will be two distinct real solutions. If it is zero, there will be exactly one real solution. If it is less than zero, there will be two complex solutions. Here, \(D=169\) which is greater than 0, implies we have two distinct real solutions for the given equation.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of the second degree. It can be expressed in the standard form as:
\[ ax^2 + bx + c = 0 \]where:

• \(a\), \(b\), and \(c\) are constants called the coefficients.
• \(x\) represents the variable or unknown that we need to solve for.
• \(a\) cannot be zero because if it were, the equation would not be quadratic but linear.

Quadratic equations often appear in various areas of mathematics and applied fields, such as physics and engineering.
To solve these equations, you might use several methods, including factoring, completing the square, or using the quadratic formula.
The quadratic formula, which arises from completing the square in the general form, allows finding the solutions by calculating the roots of the equation.

Real Solutions

Real solutions of a quadratic equation refer to the values of \(x\) for which the equation holds true with real numbers.
The number and nature of real solutions are determined by the discriminant, denoted as \(D\), which is calculated using the formula:
\[ D = b^2 - 4ac \]Once you have computed the discriminant, you can determine the solutions as follows:

• If \(D > 0\), there are two distinct real solutions. This means that the parabola representing the quadratic equation intersects the x-axis at two different points.
• If \(D = 0\), there is exactly one real solution, indicating that the parabola just touches the x-axis at one point.
• If \(D < 0\), there are no real solutions. Instead, the solutions are complex or imaginary, suggesting that the parabola does not cross the x-axis at all.

For the equation \(2x^2 + 11x - 6 = 0\) in our example, the discriminant is \(169\), which is greater than zero. Therefore, the equation has two distinct real solutions.

Coefficients

In the context of quadratic equations, coefficients are the numerical or constant factors that multiply the variables.
For a typical quadratic equation \(ax^2 + bx + c = 0\), the coefficients are:

• \(a\): The coefficient of \(x^2\), often referred to as the leading coefficient. It determines the parabola's direction of opening. If \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards.
• \(b\): The coefficient of \(x\), which affects the position and shape of the parabola.
• \(c\): The constant term, representing the y-intercept of the graph when \(x = 0\).

Understanding these coefficients is crucial when using the quadratic formula \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\),as they are essential for determining the nature of the solutions to the equation. In our original example, the coefficients identified were \(a = 2\), \(b = 11\), and \(c = -6\). This information is fundamental when calculating the discriminant and solving the equation.