Problem 28 - Algebra 1: Concepts and Skills

Question

Determine whether the equation has two solutions, one solution, or no real solution. \(3 x^{2}-6 x+3=0\)

Step-by-Step Solution

Answer

The equation has one real solution.

Step 1: Identify the coefficients

From the quadratic equation \(3 x^{2}-6 x+3=0\), we can see that \(a = 3\), \(b = -6\) and \(c = 3\).

Step 2: Calculate the discriminant

The discriminant \(D\) is calculated using the formula \(D = b^{2} - 4ac\). Insert \(-6\) for \(b\), \(3\) for \(a\), and \(3\) for \(c\), which yields \(D = (-6)^{2} - 4(3)(3) = 0\).

Step 3: Determine the number of solutions

Since the discriminant \(D = 0\), the equation has exactly one real solution.

Key Concepts

Discriminant

Quadratic equations are often represented in the form \(ax^2 + bx + c = 0\). To figure out how many real solutions these equations have, we use a special number called the discriminant.The discriminant is calculated using the formula \(D = b^2 - 4ac\). It's based on the coefficients of the equation, which we'll learn more about later.The discriminant offers a nifty way of understanding the nature of the solutions:

• If \(D > 0\), the equation has two distinct real solutions.
• If \(D = 0\), it has exactly one real solution.
• If \(D < 0\), there are no real solutions, only complex ones.

In our example, the discriminant turned out to be \(0\). This tells us that the equation \(3x^2 - 6x + 3 = 0\) has exactly one real solution. Calculating the discriminant is key when dealing with quadratic equations.

Real Solutions

Real solutions in the context of quadratic equations refer to the values of \(x\) that satisfy the equation where both the value of the solution and the components of the function are real numbers.Real solutions emerge when the discriminant \(D\) is zero or positive.Let's break down what different discriminant values mean for real solutions:

• When \(D > 0\), there are two distinct real solutions. This means that the quadratic curve crosses the x-axis at two points.
• When \(D = 0\), there is one real solution. This indicates the parabola just touches the x-axis, so there is only one point of contact.
• If \(D < 0\), the equation has no real solutions; the parabola does not touch the x-axis at all.

In our exercise, the discriminant \(D\) equals zero, so we have a scenario where the curve just touches the x-axis. This means the equation \(3x^2 - 6x + 3 = 0\) has exactly one real solution.

Coefficients

Coefficients are the numerical values placed in front of the variables in an equation. In a standard quadratic equation \(ax^2 + bx + c = 0\), the coefficients are \(a\), \(b\), and \(c\).Here's what each represents:

• \(a\) is the coefficient of the squared term \(x^2\).
• \(b\) is the coefficient of the linear term \(x\).
• \(c\) is the constant term.

The role of coefficients is crucial because they determine the shape and position of the parabola represented by the quadratic equation.In our example \(3x^2 - 6x + 3 = 0\), the coefficients are \(a = 3\), \(b = -6\), and \(c = 3\).These coefficients were important in calculating the discriminant, which helped us to determine the number of real solutions. Understanding coefficients helps in forming and solving equations efficiently.