Problem 24 - Algebra 1: Concepts and Skills

Question

Determine whether the equation has two solutions, one solution, or no real solution. \(-3 x^{2}+5 x-1=0\)

Step-by-Step Solution

Answer

The equation has two real solutions.

Step 1: Identify the coefficients

The coefficients of the quadratic equation are \(a = -3\), \(b = 5\), and \(c = -1\).

Step 2: calculating the Discriminant

Let's substitute \(a = -3\), \(b = 5\), and \(c = -1\) into the formula for the discriminant, \(b^2 - 4ac\). This gives: \( (5)^2 - 4(-3)(-1) = 25 - 12 = 13\)

Step 3: Determine the number of solutions

Since the discriminant is positive (13), according to the discriminant rule, the quadratic equation \( -3 x^{2}+5 x-1=0\) has two real solutions.

Key Concepts

Discriminant

The discriminant is a vital component in understanding quadratic equations. It provides valuable insight into the nature of the solutions for the equation. To find the discriminant, we use the formula:

• \( b^2 - 4ac \)

Here, \( b \), \( a \), and \( c \) are the coefficients from the standard form of a quadratic equation, \( ax^2 + bx + c = 0 \). The computation of this formula will help determine the type and number of solutions.
A discriminant can be positive, zero, or negative, and each gives different information:

• If the discriminant is positive, there are two distinct real solutions.
• If it is zero, there is exactly one real solution often referred to as a repeated or double root.
• If negative, there are no real solutions; instead, the solutions are complex or imaginary numbers.

Within our initial problem, the calculated discriminant is 13, which is positive, indicating two real solutions.

Real Solutions

Real solutions refer to the values of \( x \) that satisfy the quadratic equation such that the solution can be plotted on a real number line. When we discuss whether solutions are real, we're determining whether the discriminant indicates actual, feasible solutions that can be seen as intersections of the parabola represented by the quadratic equation and the x-axis.
When the discriminant is:

• Positive – the parabola intersects the x-axis at two points, meaning there are two distinct real solutions.
• Zero – the parabola touches the x-axis at exactly one point, suggesting one real solution.
• Negative – the parabola does not intersect the x-axis, thus there are no real solutions.

Having a full grasp of real solutions helps in predicting the behavior of the graph of the equation and understanding what set of solutions supports real-world scenarios.

Coefficients

In quadratic equations, coefficients are the numerical factors that multiply the variables. Every quadratic equation can be expressed in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are its coefficients. Understanding their role is crucial in solving quadratic equations:

• \( a \) is the coefficient of \( x^2 \) and should never be zero, as otherwise, it would not be a quadratic equation.
• \( b \) is the coefficient of \( x \), influencing the linear nature of the equation.
• \( c \) is the constant term, representing the y-intercept when the equation is graphed.

Each coefficient impacts the equation's discriminant and, consequently, the number and nature of its solutions. In our example, the coefficients \( a = -3 \), \( b = 5 \), and \( c = -1 \) were used to calculate the discriminant, helping to determine that the given quadratic equation has two real solutions.