Problem 81 - Algebra and Trigonometry

Question

Compute the discriminant. Then determine the number and type of solutions for the given equation. $$ x^{2}-3 x-7=0 $$

Step-by-Step Solution

Answer

The discriminant is 37, and the equation has two real and distinct solutions.

Step 1: Identifying Coefficients

In our quadratic equation \(x^{2}-3x-7=0\), the coefficients are a=1 (coefficient of \(x^{2}\)), b=-3 (coefficient of x), and c=-7.

Step 2: Calculating the Discriminant

The discriminant is calculated using the formula \(D = b^{2} - 4ac\). Substituting our values, we get \(D = (-3)^{2} - 4*1*(-7) = 9 + 28 = 37.

Step 3: Analyzing the Discriminant

The value of the discriminant is 37, which is greater than zero. This indicates that our equation has two real and distinct solutions.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation where the highest degree is 2. This means the variable, typically represented as "x," is squared. A standard form of a quadratic equation is given by \( ax^{2} + bx + c = 0 \). Here, "a," "b," and "c" represent the coefficients, and they are crucial in solving the equation. Each quadratic equation has three key components:

• The term \( ax^{2} \) is the quadratic term. The coefficient "a" must be non-zero.
• The term "bx" acts as the linear term.
• "c" is the constant term which doesn’t include the variable "x."

Understanding these components helps in determining the structure of the quadratic function. Solving a quadratic equation involves finding the values of "x" that make the equation equal to zero. For each equation, the discriminant plays a pivotal role in understanding the nature of these solutions.

Real and Distinct Solutions

For any quadratic equation, the discriminant, represented as \( D \), helps us determine the number and nature of solutions. The discriminant formula \( D = b^{2} - 4ac \) is derived from the coefficients of the quadratic equation. The value of "D" differs in indicating types of solutions:

• If \( D > 0 \), the quadratic equation has two real and distinct solutions. This means the graph of the equation intersects the x-axis at two points.
• If \( D = 0 \), it has exactly one real solution, often referred to as a repeated or double root, meaning the graph touches the x-axis at only one point.
• If \( D < 0 \), the solutions are complex or imaginary, and the graph does not intersect the x-axis.

In our original equation \( x^{2} - 3x - 7 = 0 \), the discriminant value is 37, which is greater than zero. Therefore, it confirms that there are two real and distinct solutions.

Coefficients

Coefficients are numerical values or constants that multiply the variable(s) in a polynomial equation. In a quadratic equation such as \( ax^{2} + bx + c = 0 \), the coefficients "a," "b," and "c" determine the function's shape and position on the graph.

• "a" is the coefficient of the square term. It influences the parabola's direction (opening upwards if positive and downwards if negative) and its "width."
• "b" is the coefficient of the linear term. It affects the position of the parabola along the x-axis and the parabola's axis of symmetry.
• "c" is the constant term that shifts the graph up or down along the y-axis.

In our specific equation, the values are \( a = 1 \), \( b = -3 \), and \( c = -7 \). Accurately identifying these values is essential, especially when calculating the discriminant or deriving solutions through methods like factoring or using the quadratic formula.