Problem 37 - Algebra 1

Question

Decide how many solutions the equation has. $$x^{2}-2 x-15=0$$

Step-by-Step Solution

Answer

The equation \(x^{2}-2x-15=0\) has two solutions.

Step 1: Identify the coefficients

The quadratic equation is in the form \(ax^{2}+bx+c=0\). For our equation, \(x^{2}-2x-15=0\), the coefficients are \(a=1, b=-2,\) and \(c=-15\).

Step 2: Compute the discriminant

Compute the discriminant D using the formula \(D=b^{2}-4ac\). In our case, D is \((-2)^{2} - 4*1*(-15)\) which simplifies to \(4+60=64\).

Step 3: Find the number of solutions

Because our discriminant D is positive, this means our equation \(x^{2}-2x-15=0\) has two distinct solutions.

Key Concepts

Understanding the Discriminant

A discriminant is a very important part of solving quadratic equations. It is denoted by the letter \( D \) and reveals the nature of the roots of the equation. To compute the discriminant, use the formula:

• \( D = b^2 - 4ac \)

In our example, the quadratic equation is \( x^2 - 2x - 15 = 0 \). By plugging in the values of \( a = 1 \), \( b = -2 \), and \( c = -15 \), the discriminant is calculated as follows:

• \( D = (-2)^2 - 4 \times 1 \times (-15) = 4 + 60 = 64 \)

A positive discriminant, such as 64 in this case, indicates that the quadratic equation has two distinct real solutions. So, understanding the discriminant is key to determining how many solutions you will have.

Role of Coefficients

Coefficients are the numbers in front of the variables in the quadratic equation \( ax^2 + bx + c = 0 \). These numbers define the shape and position of the parabola represented by the equation. Let's break down the coefficients in our equation \( x^2 - 2x - 15 = 0 \):

• \( a = 1 \), the coefficient of \( x^2 \), determines the direction of the parabola. Since \( a \) is positive, the parabola opens upwards.
• \( b = -2 \), the coefficient of \( x \), affects the position of the vertex and the axis of symmetry.
• \( c = -15 \), the constant term, indicates the y-intercept. This is the point where the parabola crosses the y-axis.

The coefficients are not just part of the formula; they are crucial for determining the graph's properties and solving the equation.

Determining the Number of Solutions

The number of solutions for a quadratic equation depends directly on the discriminant value (\( D \)). The discriminant helps determine the nature of the solutions:

• If \( D > 0 \), the equation has two distinct real solutions.
• If \( D = 0 \), there is exactly one real solution, also known as a repeated or double root.
• If \( D < 0 \), the equation has no real solutions and results in two complex solutions.

In our specific problem, with a discriminant of \( D = 64 \) (which is greater than zero), it tells us that the quadratic equation \( x^2 - 2x - 15 = 0 \) has two distinct real solutions. Understanding how the discriminant influences the number of solutions is essential for solving quadratic equations effectively.