Problem 43 - Elementary Algebra

Question

Classify each of the equations for the following problems by degree. If the term linear, quadratic, or cubic applies, state it. $$ x^{2}+x-4=7 x^{2}-2 x+9 $$

Step-by-Step Solution

Answer

Question: Based on the step-by-step solution, classify the given equation $$x^2 + x - 4 = 7x^2 - 2x + 9$$ by degree and state whether it is linear, quadratic, or cubic. Answer: The given equation is quadratic.

Step 1: Combine like terms

First, we need to move all the terms to one side of the equation. Subtract $$7x^2$$ and $$-2x$$ from both sides of the equation to get: $$ x^2 + x - 4 - 7x^2 + 2x - 9 = 0 $$ Now, combine the like terms on the left side of the equation: $$ -x^2 + 3x - 13 = 0 $$

Step 2: Determine the degree of the polynomial

Now that we have combined the like terms, we can see that the given equation is in the form of a polynomial. To classify the equation by degree, we need to find the highest exponent of the variable x in the polynomial. In this case, the highest exponent is 2 (from the term $$-x^2$$).

Step 3: Classify the polynomial by degree

Since the highest exponent is 2, the polynomial is classified as a quadratic equation. We can verify this by noting that quadratic equations follow the general form$$ax^2+bx+c=0$$, where a, b, and c are constants, and a is non-zero. Our simplified equation$$-x^2 + 3x - 13 = 0$$matches this form. So, the given equation is quadratic.

Key Concepts

Polynomial Degree

When we talk about the degree of a polynomial, we're referring to the highest power of the variable within an equation. It's a fundamental concept for understanding different types of polynomials and their behavior. The degree tells us much about a polynomial's characteristics, such as the number of roots it can have and the shape of its graph. For instance, a first-degree polynomial, like y = 2x + 3, is a straight line, while second-degree polynomials, which we'll discuss further in the quadratic equation section, create parabolic graphs.

To find the degree of a polynomial, simply identify the highest exponent of the variable in the expression after combining like terms. If you have the equation x^3 - 2x^2 + 4x - 1, the degree is three because the term with the highest exponent is x^3. Remember, constants, like the number 5, are considered polynomials of degree zero, and the absence of a variable, signifying x^0, also equates to a degree of zero.

Quadratic Equation

A quadratic equation is a second-degree polynomial that takes the form ax^2 + bx + c = 0, where a, b, and c are constants with a ≠ 0. The graphs produced by quadratic equations are parabolas, which can open either upwards or downwards depending on the sign of a. These parabolas also have a vertex, which is the maximum or minimum point of the curve.

Quadratic equations are essential in many areas of algebra and appear frequently in various mathematical applications, from physics to finance. Solving a quadratic equation can involve factoring, completing the square, using the quadratic formula, or even graphing. The importance of identifying a quadratic equation lies in the specific set of solutions it can offer, typically two distinct real or complex roots.

Combining Like Terms

Combining like terms is a critical skill in algebra that simplifies expressions and makes it possible to solve equations efficiently. Like terms are terms in an expression that have the same variable raised to the same power. By combining them, we essentially add or subtract their coefficients. This process often makes the solution to equations much clearer.

For example, if we have the expression 4x - 2x^2 + 3x + x^2, we can combine like terms to simplify it to 7x - x^2. Notice how we combined 4x and 3x because they are both the first power of x, and we also combined -2x^2 and x^2 since they are both the second power of x. Correctly combining like terms is a foundational step for solving equations and should be done with care to ensure all terms are accurately accounted for.