Problem 77 - College Algebra

Question

In Exercises \(75-82\), compute the discriminant. Then determine the number and type of solutions for the given equation. $$2 x^{2}-11 x+3=0$$

Step-by-Step Solution

Answer

The discriminant is 97, and the equation has two distinct real solutions.

Step 1: Identify the coefficients

The quadratic equation is in the standard form \(ax^{2} + bx + c = 0\). From the equation \(2x^{2}-11x+3=0\), the coefficients are \(a = 2\), \(b = -11\), and \(c = 3\).

Step 2: Calculate the discriminant

The discriminant \(D\) is given by the formula \(D = b^{2} - 4ac\). Substituting the identified coefficients into this formula gives \(D = (-11)^{2} - 4*2*3 = 121 - 24 = 97\)

Step 3: Determine the number and type of solutions

Since \(D > 0\), the equation has two distinct real roots.

Key Concepts

Coefficients of a Quadratic Equation

In a quadratic equation of the standard form, which is expressed as \( ax^2 + bx + c = 0 \), the constants \( a \), \( b \), and \( c \) are referred to as the coefficients. Here, \( a \) is the coefficient of the squared term (\( x^2 \)), \( b \) is the coefficient of the linear term (\( x \)), and \( c \) represents the constant term. The values of these coefficients play a critical role in determining the characteristics of the equation's graph as well as the nature of its solutions.

For the quadratic equation \( 2x^2 - 11x + 3 = 0 \), the coefficient \( a \) is 2, representing how the parabola opens and its width. The coefficient \( b \) is -11, impacting the location of the vertex of the parabola on the x-axis. Lastly, \( c \) is 3, which affects the position of the parabola on the y-axis.

Calculating the Discriminant

The discriminant of a quadratic equation is a value that can be calculated using the coefficients \( a \), \( b \), and \( c \). It is represented by the formula \( D = b^2 - 4ac \). The discriminant is notably important because it tells us about the nature of the roots without actually solving the equation.

For example, in the equation \( 2x^2 - 11x + 3 = 0 \), to calculate the discriminant, you plug the values of \( a \), \( b \), and \( c \) into the discriminant formula, resulting in \( D = (-11)^2 - 4(2)(3) \) which simplifies to \( D = 121 - 24 = 97 \). This discriminant value is positive, indicating that there will be two distinct real solutions to the quadratic equation.

Determining Number and Type of Solutions

Once you have the discriminant, you can determine both the number and the type of roots (solutions) the quadratic equation will have. If the discriminant \( D > 0 \), the quadratic equation has two distinct real roots. If \( D = 0 \), the equation has exactly one real root, meaning it touches the x-axis at a single point. Alternatively, if \( D < 0 \), there are no real roots, and instead, there are two complex roots.

For our equation \( 2x^2 - 11x + 3 = 0 \), with discriminant \( D = 97 \), which is greater than zero, we can conclude that there are two distinct real solutions to the equation. These solutions can be found by further solving the equation using methods such as factoring, completing the square, or applying the quadratic formula.