Problem 69 - Algebra 1: Concepts and Skills

Question

Which quadratic equation is written in standard form? A. \(8 x+5 x^{2}-9=0\) B. \(5 x^{2}+8 x=9\) C. \(5 x^{2}+8 x-9=0\) D. \(9-8 x-5 x^{2}=0\)

Step-by-Step Solution

Answer

The quadratic equation that is written in standard form is C. \(5 x^{2}+8 x-9=0\)

Step 1: Review all the quadratics

Look at all the given quadratic equations. The equations are: A. \(8 x+5 x^{2}-9=0\) B. \(5 x^{2}+8 x=9\) C. \(5 x^{2}+8 x-9=0\) D. \(9-8 x-5 x^{2}=0\)

Step 2: Compare with the standard form

Compare the equations given with the standard form of the quadratic equation, \(ax^{2} + bx + c = 0\). It is obvious that option A and D do not follow the standard form since the term \(ax^{2}\) is not the first term. Option B is not in the standard form since the equation is not equal to zero.

Step 3: Identify the correct form

After reviewing all the equations, it can be deduced that option C, \(5 x^{2}+8 x-9=0\) is the only equation that is in standard form of a quadratic equation. The standard form includes the quadratic term first, followed by the linear term and the constant set equal to zero. Option C is structured identically.

Key Concepts

Quadratic Equations

Quadratic equations are foundational elements in high school algebra that every student encounters. A quadratic equation is an algebraic expression of the second degree, which means it includes a term with a variable raised to the power of two. The general standard form of a quadratic equation is represented as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero, as this would make it a linear equation, not quadratic.

Understanding the structure of the standard form is crucial because it allows students to apply various methods for finding the solutions, or roots, of the quadratic equation. These methods include factoring, completing the square, and using the quadratic formula. Recognize that any quadratic equation can be reorganized into standard form by simple algebraic manipulation - rearranging and combining like terms.

It's important to identify the standard form to correctly apply these solution methods, which is why comparing given equations to the standard form, as shown in the exercise, is a necessary skill in algebra.

Algebraic Expressions

An algebraic expression refers to a mathematical phrase that can include numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. It becomes particularly interesting when dealing with polynomials, which include terms of variables raised to different powers.

In the context of quadratic equations, the algebraic expressions consist of three major parts: the quadratic term \(ax^2\), the linear term \(bx\), and the constant term \(c\). To form a quadratic equation, these parts are combined and set equal to zero. Understanding how to manipulate these expressions is a core aspect of high school algebra, enabling students to solve a wide range of problems.

Key Features of Algebraic Expressions

• Terms are the building blocks of algebraic expressions.
• Coefficients are the numerical factor of terms with variables.
• Like terms can be combined or simplified by addition or subtraction.

The capability to recognize and simplify algebraic expressions is not only vital in solving quadratic equations but also forms the basis for more complex algebraic functions and calculus.

High School Algebra

High school algebra is more than just an academic requirement; it's a critical thinking tool that aids in solving real-world problems. Central to this subject are concepts like solving equations, understanding functions, and analyzing graphical representations.

Quadratic equations represent an important topic within high school algebra, as they appear frequently both in further mathematics courses and in various application areas such as physics and engineering. The skills developed from understanding quadratic equations, such as critical thinking, problem-solving, and logical reasoning, are universally valuable.

Why Learning Algebra Matters

• It helps to develop logical thinking and problem-solving skills.
• It is essential for understanding advanced mathematics and sciences.
• It has practical applications in various fields like technology, healthcare, and finance.

Students should approach algebra not just as a hurdle to clear but as a fundamental language of the modern world, describing patterns and relationships that help decode the complexities of various phenomena.