Problem 6 - Elementary Algebra

Question

Which of the following equations are quadratic equations? Answer "yes" or "no" to each equation. $$ 9 x^{2}-2 x+6=4 x^{2}+8 $$

Step-by-Step Solution

Answer

Answer: Yes

Step 1: Simplify the given equation

To simplify the equation, we need to move all the terms to one side. Subtract $$ 4x^2 $$ from both sides, and subtract 8 from both sides. $$ 9x^2 - 2x + 6 - 4x^2 - 8 = 0 $$

Step 2: Combine like terms

Now, combine the like terms in the equation: $$ (9x^2 - 4x^2) + (-2x) + (6 - 8) = 0 $$

Step 3: Write the simplified equation

Our simplified equation becomes: $$ 5x^2 - 2x - 2 = 0 $$

Step 4: Check if it's a quadratic equation

The simplified equation is of the form $$ ax^2 + bx + c = 0 $$ with a=5, b=-2, and c=-2, and $$a \neq 0$$, so it is a quadratic equation. Answer: Yes

Key Concepts

Simplifying Equations

Simplifying equations is a crucial step to understanding their true form, especially when dealing with quadratic equations. The key to simplifying is to rearrange the terms such that all expressions are moved to one side of the equation and set equal to zero. This makes it much easier to evaluate and solve.
To begin simplifying the equation \(9x^2 - 2x + 6 = 4x^2 + 8\), you should perform the following steps:

• First, subtract \(4x^2\) from both sides to eliminate it from the right side.
• Next, subtract 8 from both sides to move constant terms together.
• The goal is to have all terms on one side of the equation, resulting in \(9x^2 - 2x + 6 - 4x^2 - 8 = 0\).

Remember, the purpose of this rearrangement is to aid in identifying and handling the terms more effectively in subsequent steps.

Combining Like Terms

The process of combining like terms is essential to reducing equations to a simpler form. After moving all terms from one side of the equation to the other, the next step is to combine any terms that are similar. This means:

• Identifying terms with the same variable and power (e.g., \(x^2\) terms).
• Adding or subtracting these coefficients as needed.

In our example, after you have \(9x^2 - 2x + 6 - 4x^2 - 8 = 0\), you combine the \(x^2\) terms:

• \(9x^2 - 4x^2\) becomes \(5x^2\).

For the constant terms and first-degree \(x\) terms:

• The constant terms \(6 - 8\) become -2.
• There is only one \(-2x\) term, which remains unchanged.

This results in a simplified equation \(5x^2 - 2x - 2 = 0\), ready to check for quadratic form.

Checking Quadratic Form

To determine if an equation is quadratic, we need to verify that it fits the standard form \(ax^2 + bx + c = 0\). This form expresses a polynomial equation of degree two, characterized by:

• An \(x^2\) term (where the coefficient \(a\) is not zero).
• A possible \(x\) term (denoted as \(bx\)).
• A constant term \(c\).

In our simplified equation \(5x^2 - 2x - 2 = 0\):

• \(a = 5\), \(b = -2\), and \(c = -2\).
• Importantly, \(a\) is not zero, confirming that it's quadratic.

Thus, this is indeed a quadratic equation, as it matches the structure required for one. Always ensure that the coefficient \(a\) is not zero, as this is the defining characteristic of a quadratic equation.