Problem 20 - Algebra 1: Concepts and Skills

Question

Find the discriminant of the quadratic equation. \(2 x=x^{2}-x\)

Step-by-Step Solution

Answer

The discriminant of the quadratic equation is 1.

Step 1: Rearrange the Equation

First, rearrange the given equation \(2x = x^{2} - x\) into standard form \(ax^{2} + bx + c\). This is done by moving all terms to one side of the equation. This gives us \(x^{2} - 3x + 2 = 0\).

Step 2: Identify Coefficients

From our quadratic equation \(x^{2} - 3x + 2 = 0\), identify the values of a, b, and c. In this case, \(a = 1\), \(b = -3\), and \(c = 2\). These values will be used to calculate the discriminant.

Step 3: Calculate the Discriminant

The discriminant of a quadratic equation is calculated as \(b^{2} - 4ac\). Substitute the identified coefficients into this formula to find the discriminant. So, \((-3)^{2} - 4*1*2 = 9 - 8 = 1\)

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree two, which means it involves a term with the variable raised to the exponent of two. Typically, a quadratic equation can be expressed in the form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are known as coefficients, and \( x \) represents the unknown variable we are trying to solve for.

Quadratic equations are fundamental in mathematics because they come up in various scenarios, like physics problems, financial calculations, and even computer graphics. Solving these equations typically involves finding the values of \( x \) that satisfy the equation. These values are often called the "roots" or "solutions" of the equation.

• The term with \( x^2 \) is called the quadratic term.
• The term with \( x \) is known as the linear term.
• The constant term in the equation is \( c \).

Understanding how to manipulate and solve a quadratic equation is crucial for progressing in math.

Standard Form

The standard form of a quadratic equation is essential for solving and analyzing the equation. Rearranging an equation into the standard form \( ax^2 + bx + c = 0 \) makes it easier to understand and apply various solution methods, such as factoring or using the quadratic formula. This format highlights the coefficients, which play a pivotal role in solving the equation.

To convert any quadratic expression into standard form:

• Ensure that all terms are on one side of the equation, leaving zero on the other. This might involve moving terms across the equal sign.
• Arrange the terms in descending order of their exponent values.
• Make sure to express the coefficients as clearly identified numerical values.

For the equation provided in our exercise, \( 2x = x^2 - x \), rearranging gives \( x^2 - 3x + 2 = 0 \) in standard form.

Coefficients

Coefficients are the numbers that multiply the variables or terms in an equation. In the context of quadratic equations, these numbers are referred to as \( a \), \( b \), and \( c \), each corresponding to the respective terms in the standard form equation \( ax^2 + bx + c = 0 \). Identifying these coefficients correctly is crucial as they are used in calculating important characteristics of the quadratic equation, such as the discriminant and the solutions.

• \( a \) is the coefficient of the \( x^2 \) term. It indicates the degree and direction of the parabola's opening.
• \( b \) is the coefficient of the \( x \) term, playing a role in determining the vertex of the parabola.
• \( c \) is the constant term, which affects the vertical position of the parabola.

In the given equation \( x^2 - 3x + 2 = 0 \), \( a = 1 \), \( b = -3 \), and \( c = 2 \). These coefficients are used in the discriminant formula \( b^2 - 4ac \), which helps determine the nature of the roots of the quadratic equation.