Problem 72 - Algebra 1: Concepts and Skills

Question

Write the equation in standard form. $$ 12 x=x^{2}+25 $$

Step-by-Step Solution

Answer

The equation in standard form is \(x^2 - 12x + 25 = 0\).

Step 1: Rearrange the Equation

First need to draw all the terms on one side of the equation. This is done by subtracting \(12x\) from both sides of the equation. This gives: \(x^2 - 12x + 25 = 0\).

Step 2: Verify the Order

Check that the equation is in descending order of exponents. From the equation \(x^2 - 12x + 25 = 0\), it is clear that it's already in descending order.

Key Concepts

Understanding Quadratic Equations

A quadratic equation is a type of polynomial equation that involves an unknown variable raised to the power of two. Its general form can be expressed as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) cannot be zero. This is because if \( a \) were zero, the equation wouldn't be quadratic, but linear instead.
To identify a quadratic equation, look for the highest power of the variable, usually denoted as \( x^2 \). The presence of this squared term is what characterizes the equation as quadratic.
Quadratic equations are fundamental in algebra and appear in various real-life applications such as physics, engineering, and economics, where they are used to model parabolic shapes and calculate optimal values.

Rearranging Equations

Rearranging equations involves altering the format of an equation without changing its solutions. When we talk about putting an equation into standard form, we're typically aiming for a more organized presentation that makes it easier to identify important features like the quadratic term.
In the given exercise, you start with the equation \(12x = x^2 + 25\). To rearrange it into standard form, you want to move all terms to one side, resulting in \( x^2 - 12x + 25 = 0 \).
Moving terms involves using basic algebraic operations such as addition and subtraction. In this case, subtract \( 12x \) from both sides. This step helps in creating a zero on one side of the equation, which is a crucial aspect when solving quadratics either by factoring, completing the square, or using the quadratic formula.

Importance of Descending Order of Exponents

When writing a polynomial equation, particularly in standard form, it's important to arrange terms in descending order of exponents. This means starting with the highest power of the variable and proceeding to the lower ones. For example, in \( x^2 - 12x + 25 = 0 \), 'descending order of exponents' is perfectly maintained as the terms are arranged from \( x^2 \) to \( x \) and finally, the constant term 25.
The descending order makes equations easier to analyze and solve, as it helps learners quickly identify the leading coefficient and the degree of the polynomial. These are key to understanding the behavior of the graph of the equation and its solutions.
Therefore, whenever you handle equations, especially quadratic ones, always check if the terms are organized in descending order. It's a simple step but crucial for clarity and effective problem solving.