Problem 10 - College Algebra and Calculus: An Applied Approach

Question

Write the quadratic equation in general form. $$ \frac{x^{2}-7}{3}=2 x $$

Step-by-Step Solution

Answer

The quadratic equation in general form will be \(x^{2} - 6x - 7 = 0\)

Step 1: Multiply Both Sides to Eliminate the Fraction

To make the equation simpler to work with, multiply both sides by 3. This results in \(3*\frac{x^{2}-7}{3}=3*2x\), which simplifies to \(x^2 - 7 = 6x\).

Step 2: Rearrange the Equation into Standard Form

In standard form, the quadratic equation should look like \(ax^{2} + bx + c = 0\). This can be achieved by rearranging the equation from step 1. We should subtract \(6x\) from both sides of the equation, resulting in \(x^2 - 6x - 7 = 0\)

Step 3: Verify You Have The Correct Form

To ensure the equation is now in the correct form \(ax^{2} + bx + c = 0\), check to see that all terms are on one side of the equation while leading to zero (0) on the other side.

Key Concepts

Solving Quadratic Equations

Quadratic equations are a foundational concept in algebra. A quadratic equation is typically in the form of \(ax^{2} + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable. Solving these equations may seem intimidating at first, but with a systematic approach, it can become quite manageable.

The process begins by ensuring that the equation is in standard form, as the exercise shows. Sometimes, this involves simple algebraic operations such as multiplying both sides to eliminate fractions or adding and subtracting terms to one side. After doing that, the quadratic equation is then ready to be solved by different methods, such as factoring, using the quadratic formula, completing the square, or graphing. Each of these methods can be applied depending on the context of the problem and the numbers involved.

Factoring

When the quadratic equation can be factored, the solution involves finding two binomials that multiply to give the quadratic term and the constant term, while adding up to the middle term. However, factoring is not always possible for all quadratic equations.

Completing the Square

Completing the square is a method that involves rearranging and adding a constant to both sides of the equation, so that the left-hand side becomes a perfect square trinomial.

Graphing

Another way to solve a quadratic equation is by graphing it and finding the points where the graph intersects with the x-axis. These intersection points are the solutions to the equation.

Algebraic Operations

Algebraic operations are the bedrock of solving quadratic equations. These operations include addition, subtraction, multiplication, and division. They are used to manipulate the equation into a workable form that allows the application of various techniques to find the solution.

As seen in the exercise, to eliminate the fraction, we multiply both sides of the equation, which is one of the basic algebraic operations. It is crucial to maintain the balance of the equation by applying the same operation to both sides. In the case of quadratic equations, we often need to move terms around to get them into the standard form. This often involves subtracting or adding terms across the equal sign or factoring.

Understanding how to correctly apply these operations is essential as it is a skill that will be used repeatedly not only in algebra but throughout different areas of mathematics. Practice is vital, and recognizing when and how to employ each operation comes with experience.

Quadratic Formula

When other methods are not applicable or would be overly complicated, the quadratic formula provides a straightforward solution to quadratic equations. The quadratic formula is \(x = \frac{{-b \pm \sqrt{b^2 - 4ac}}}{{2a}}\). It's derived from the process of completing the square on a quadratic equation and can solve for \(x\) in any quadratic equation that can be rearranged into the standard form of \(ax^2 + bx + c = 0\).

The beauty of the quadratic formula is its universality. It does not matter if the quadratic equation can be factored or not; the formula will still work. The discriminant, \(b^2 - 4ac\), within the quadratic formula is particularly important because it determines the nature and number of solutions. If it's positive, there are two real solutions; if zero, there is one real solution; and if negative, there are no real solutions—only complex ones.

Using the quadratic formula can sometimes be more efficient than other methods, and having a solid understanding of this formula is a valuable tool for any student studying algebra.