Problem 70 - Algebra 1: Concepts and Skills

Question

Write the equation in standard form. $$ 9-6 x=2 x^{2} $$

Step-by-Step Solution

Answer

The equation in standard form is \(x^{2} + 3x - 4.5 = 0\).

Step 1: Set Up the Quadratic in Standard Form

To get the equation into standard form, reorganize the equation such that the right side is equal to zero. This can be done by subtracting \(9-6x\) from both sides to get: \[2x^{2} + 6x - 9 = 0\]

Step 2: Simplification

Here, we observe that all of the terms of our equation: \(2x^{2} + 6x - 9\), are divisible by 2. Dividing the whole equation by 2 gives: \[x^{2} + 3x - 4.5 = 0\]. The equation is now in standard form.

Key Concepts

Standard Form of a Quadratic Equation

When we talk about the standard form of a quadratic equation, we refer to a specific arrangement of the terms. A quadratic equation generally takes the form:

• \[ ax^2 + bx + c = 0 \]

where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable.
This arrangement makes solving the equation manageable, as it prepares the equation for methods such as factoring, using the quadratic formula, or completing the square.
In the original exercise, the given equation was \(9 - 6x = 2x^2\). The goal was to rewrite it so the terms appear in the standard form, with everything on one side and zero on the other. This rearrangement helps systematically isolate the variable and find solutions to the equation.

Simplifying Equations

Simplifying equations is a crucial step in solving them effectively. By simplifying, you reduce complexity and make the equation easier to work with.
Once the equation \(2x^2 + 6x - 9 = 0\) was organized into standard form, the next step was to simplify where possible.
In this context, simplifying involved dividing the entire equation by the greatest common factor (GCF) shared amongst the coefficients.

• This means dividing each term by \(2\), the GCF of the equation, resulting in the simplified form \(x^2 + 3x - 4.5 = 0\).

Simplification helps to streamline the equation, making subsequent operations like factoring or graphing more straightforward. It is essential because it can reduce large coefficients that might be unwieldy or lead to errors in calculation.

Understanding Polynomial Equations

Polynomial equations are expressions that consist of variables and coefficients, combined using addition, subtraction, and multiplication.
In mathematics, the degree of the polynomial is determined by the highest power of the variable in the expression. A quadratic equation is a specific type of polynomial where the highest exponent is \(2\).

• For example, \(x^2 + 3x - 4.5 = 0\) is a polynomial equation of degree 2.

Recognizing the form and degree of polynomial equations helps in choosing appropriate methods for solving them.
Such equations could represent a variety of real-world phenomena, from projectile motion to economic models.
Understanding how to manipulate their form allows us to solve them effectively, paving the way for deeper mathematical insights.