Problem 22 - Elementary Algebra

Question

For the following problems, write the values of \(a, b,\) and \(c\) in quadratic equations. $$ 7 x^{2}+2 x+1=6 x^{2}+x-9 $$

Step-by-Step Solution

Answer

Answer: \(a = 1\), \(b = 1\), and \(c = 10\)

Step 1: Rewrite the given equation in the standard form

To rewrite the given equation in the standard form, we should move all the terms to one side of the equation. $$ 7x^2 + 2x + 1 - (6x^2 + x - 9) = 0 $$

Step 2: Simplify the equation

Now, let's simplify the equation by combining like terms. $$ (7x^2 - 6x^2) + (2x - x) + (1 + 9) = 0 $$ $$ x^2 + x + 10 = 0 $$

Step 3: Identify the values of \(a\), \(b\), and \(c\)

Comparing the simplified equation with the standard form of a quadratic equation \(ax^2+bx+c=0\), we can identify the values of \(a\), \(b\), and \(c\) as follows: $$ a = 1 , \quad b = 1 , \quad c = 10 $$

Key Concepts

Standard Form of Quadratic Equations

Quadratic equations can often appear in various forms, but there's a very neat version called the *standard form*. To express a quadratic equation in this manner, we need it in the format \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, and \( x \) represents a variable.

• \( a \) is the coefficient of \( x^2 \),
• \( b \) is the coefficient of \( x \), and
• \( c \) is the constant term.

In order to get an equation into this form, you need to have all the terms involving the variable \( x \) on one side of the equation and zero on the other side. This is why we often start by moving all terms to one side when dealing with a quadratic equation.
For example, if we had an equation like \( 7x^2 + 2x + 1 = 6x^2 + x - 9 \), the first step to getting it into standard form would involve moving all terms to one side, like this: \( 7x^2 + 2x + 1 - (6x^2 + x - 9) = 0 \). Notice how everything involving \( x \) is now on the left side, set against zero on the right.

Simplifying Equations

Simplifying equations involves reducing them to their simplest form. This means combining all the terms that can be summed or subtracted together. This makes it easier to work with and understand the equation. In the process of conversion to standard form, we aim to simplify terms as much as possible.

Steps to Simplify

• First, distribute any negative signs across parentheses.
• Next, add or subtract coefficients of like terms.

Let's take our ongoing example from earlier: now it looks like \((7x^2 + 2x + 1) - (6x^2 + x - 9) = 0\). When you distribute the negative sign within the brackets, the equation becomes \( 7x^2 + 2x + 1 - 6x^2 - x + 9 = 0 \). The next step is to combine like terms: \( (7x^2 - 6x^2) + (2x - x) + (1 + 9) = 0 \), which simplifies down to \( x^2 + x + 10 = 0 \).
Simplifying equations sufficiently can make it easier to apply further methods for solving quadratic equations, such as factoring or using the quadratic formula.

Combining Like Terms

When simplifying equations, identifying and combining like terms is crucial. *Like terms* are terms that have the same variables raised to the same power. In our journey to simplify and solve the equation, this concept is repeatedly used.
For instance, in an expression like \( 7x^2 - 6x^2 + 2x - x + 1 + 9 \), we identify:

• "\( 7x^2 \)" and "\( -6x^2 \)" as like terms because they both contain the variable \( x \) raised to the power of 2.
• Similarly, "\( 2x \)" and "\( -x \)" are like terms since they both have the variable \( x \).

After identifying like terms, combine them simply by performing the arithmetic operation indicated. Here’s how it works:

• Combine \( 7x^2 \) and \(-6x^2\) to get \( x^2 \).
• Combine \( 2x \) and \( -x \) to arrive at \( x \).
• Lastly, combine the constants 1 and 9 to make 10.

After combining, the equation \( 7x^2 + 2x + 1 - 6x^2 - x + 9 = 0 \) simplifies down to \( x^2 + x + 10 = 0 \). Understanding and mastering the concept of combining like terms is crucial in solving algebraic equations efficiently.