Problem 4 - Algebra 1

Question

Find the discriminant for the equation. Then tell if the equation has two solutions, one solution, or no real solution. $$-3 x^{2}+6 x-3=0$$

Step-by-Step Solution

Answer

The given equation has one real solution.

Step 1: Identify the coefficients

The coefficients \(a\), \(b\), and \(c\) in the equation \(-3x^{2} + 6x - 3 = 0\) are -3, 6, and -3 respectively.

Step 2: Calculate the discriminant

Use the formula \(D = b^2 - 4ac\) to calculate the discriminant. Insert the coefficients into the formula to get \(D = (6)^2 - 4*(-3)*(-3) = 36 - 36 = 0\).

Step 3: Determine the number of solutions

The value of the discriminant, \(D = 0\), which means there is exactly one real solution for the quadratic equation.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that involves variables raised to the power of two at maximum. It is generally written in the form:\[ ax^2 + bx + c = 0 \]where:

• \(a\), \(b\), and \(c\) are constants known as coefficients.
• \(x\) represents the unknown variable.

The standard quadratic equation has a degree of two, which is the highest power of the variable present. This degree dictates that there can be up to two solutions for any given quadratic equation. These solutions are the values of \(x\) that satisfy the equation when replaced into it. The nature and number of solutions can be inferred using the discriminant, which is derived from the equation’s coefficients.

Real Solution

When discussing real solutions to quadratic equations, we focus on the values of \(x\) that satisfy the equation and are real numbers.
Real numbers include all rational and irrational numbers, essentially any number that can exist on the number line.
For quadratic equations, the discriminant \(D\) plays a critical role in determining whether these solutions exist.Here's what the discriminant tells us:

• If \(D > 0\), there are two distinct real solutions.
• If \(D = 0\), there is exactly one real solution, also called a repeated or double root.
• If \(D < 0\), there are no real solutions, indicating instead the presence of complex solutions.

Understanding the discriminant helps predict the type and number of solutions without solving the equation directly.

Coefficients

Coefficients are the numerical factors in terms related to variables in a quadratic equation. In the general form of the quadratic equation \(ax^2 + bx + c = 0\), each term's coefficient affects the equation in different ways:

• \(a\): The quadratic coefficient, it affects the parabola's direction (up or down) and width.
• \(b\): The linear coefficient, affects the parabola's shape and vertex location on the x-axis.
• \(c\): The constant term, affects where the parabola intersects the y-axis.

The coefficients are integral in calculating the discriminant \(b^2 - 4ac\), which helps determine the nature of the solutions. Recognizing and accurately identifying these coefficients is essential in solving and understanding quadratic equations. This understanding allows for effective predictions regarding the solutions' nature through the discriminant.