Problem 80 - Algebra 1: Concepts and Skills

Question

Tell whether the equation has two solutions, one solution, or no real solution. $$ 2 x^{2}-3 x-1=0 $$

Step-by-Step Solution

Answer

The given equation has two real solutions.

Step 1: Identify the values of a, b and c

Here the quadratic equation is \( 2x^{2} - 3x - 1 = 0 \). So comparing this with the standard quadratic equation \( ax^{2} + bx + c = 0 \), it can be identified that \(a = 2\), \(b = -3\), and \(c = -1\).

Step 2: Calculate the Discriminant

The nature of the roots of a quadratic equation is determined by the discriminant. The formula for the discriminant is \( D = b^{2} - 4ac \). Substituting the values of a, b and c into the formula gives \( D = (-3)^{2} - 4 * 2 * (-1) = 9 + 8 = 17 \).

Step 3: Determine the Number of Solutions

Since the discriminant is greater than 0 (D = 17 > 0 ), there are two distinct real solutions to the quadratic equation.

Key Concepts

Understanding the Discriminant

The discriminant in a quadratic equation is a key factor in understanding the nature of its solutions. In the quadratic equation given as \( ax^2 + bx + c = 0 \), the discriminant \( D \) is defined by the formula \( D = b^2 - 4ac \). This simple calculation provides a wealth of information about the roots of the equation.

• If \( D > 0 \), the equation has two distinct real solutions.
• If \( D = 0 \), the equation has exactly one real solution, also known as a repeated or double root.
• If \( D < 0 \), the equation has no real solutions, meaning the roots are complex or imaginary numbers.

Determining the discriminant is often the first step in solving a quadratic equation, as it tells us how many solutions to expect and the nature of these solutions. In our provided problem, we found \( D = 17 \), which indicates that there are two distinct real solutions.

Exploring the Number of Solutions

The number of solutions to a quadratic equation depends on the discriminant. As noted above, the discriminant lets us know whether there are two solutions, one solution, or none. It's like a road sign guiding us on what to expect.

When the discriminant is positive (\( D > 0 \)), as in our example where \( D = 17 \), it means the quadratic graph crosses the x-axis at two distinct points. Hence, there are two distinct solutions to the equation.

In contrast, a zero discriminant \( (D = 0) \) results in a single solution because the graph just touches the x-axis at one point, indicating a perfect square trinomial. Finally, a negative discriminant \( (D < 0) \) tells us there are no real solutions since the graph does not intersect the x-axis at all.

Identifying Real Solutions

Real solutions are the x-values where the quadratic equation equals zero, meaning where the graph intersects the x-axis. The discriminant plays a crucial role in revealing whether these solutions are real or not.

• When \( D > 0 \), we have two real solutions, as the parabola crosses the x-axis twice.
• When \( D = 0 \), we have one real solution, because it only touches the x-axis once.
• When \( D < 0 \), there are no real solutions; instead, the solutions are complex, and the graph never meets the x-axis.

Real solutions are necessary when dealing with problems that require actual feasible answers like those seen in physics or engineering. In real-world applications, having two distinct real solutions offers more versatility in solving actual scenarios where one might need different potential outcomes.