Problem 2 - Beginning and Intermediate Algebra

Question

Which of the following are quadratic equations? A. \(x+2 y=0\) B. \(x^{2}-8 x+16=0\) C. \(2 x^{2}-5 x=3\) D. \(x^{3}+x^{2}+4=0\)

Step-by-Step Solution

Answer

B and C are quadratic equations.

Step 1: Identify Standard Form of a Quadratic Equation

A quadratic equation is typically in the standard form: \[ ax^2 + bx + c = 0 \] where \(a, b,\) and \(c\) are constants with \(a eq 0\).

Step 2: Analyze Option A

The equation is given as: \[ x + 2y = 0 \] This equation does not have a term with \(x^2\). Therefore, it is not a quadratic equation.

Step 3: Analyze Option B

The equation is given as: \[ x^2 - 8x + 16 = 0 \] This equation fits the standard form of a quadratic equation \(ax^2 + bx + c = 0\) with \(a = 1, b = -8,\) and \(c = 16\). Therefore, it is a quadratic equation.

Step 4: Analyze Option C

The equation is given as: \[ 2x^2 - 5x = 3 \] Rewriting it in standard form, subtract 3 from both sides: \[ 2x^2 - 5x - 3 = 0 \] This equation fits the standard form with \(a = 2, b = -5,\) and \(c = -3\). Therefore, it is a quadratic equation.

Step 5: Analyze Option D

The equation is given as: \[ x^3 + x^2 + 4 = 0 \] This equation has a term with \(x^3\) which makes it a cubic equation, not a quadratic equation.

Key Concepts

standard form

To determine if an equation is a quadratic equation, we typically need it to be in the standard form. The standard form for a quadratic equation is: ax^2 + bx + c = 0 where 'a', 'b', and 'c' are constants, and 'a' cannot be zero. If an equation can be rearranged into this form, it is considered quadratic. For example, if you're given the equation x^2 - 8x + 16 = 0, you can see it fits the form with a=1, b=-8, and c=16.

constants

In algebra, constants refer to values that do not change. In a quadratic equation written in standard form: ax^2 + bx + c = 0 'a', 'b', and 'c' are constants. Constants are crucial because they determine the shape and position of the parabola represented by a quadratic equation. In the equation 2x^2 - 5x - 3 = 0, 2 is the constant 'a', -5 is the constant 'b', and -3 is the constant 'c'. Each constant plays a significant role in defining the properties of the quadratic equation.

cubic equations

Sometimes, equations might not fit the quadratic form. For example, the equation x^3 + x^2 + 4 = 0 includes a term with x^3, making it a cubic equation, not a quadratic one. Cubic equations are in the form: ax^3 + bx^2 + cx + d = 0 Here, 'a', 'b', 'c', and 'd' are constants just like in the quadratic case, but 'a' must again be non-zero. The presence of the x^3 term makes the analysis and solutions of cubic equations more complex compared to quadratic ones. Therefore, x^3 + x^2 + 4 = 0 is a cubic equation since it has an x^3 term.

algebraic expressions

Understanding algebraic expressions is fundamental to manipulating and solving equations. An algebraic expression is a combination of variables and constants interconnected by operations such as addition, subtraction, multiplication, and division. For instance: x^2 - 8x + 16 This is a quadratic algebraic expression. When set to zero, it becomes an equation: x^2 - 8x + 16 = 0. This transformation from expression to equation allows us to solve for the values of x. By comprehending these expressions, one can rearrange and identify the specific forms like quadratic and cubic, aiding in categorization and solution of problems.