Problem 37 - College Algebra and Calculus: An Applied Approach

Question

In Exercises 37-46, solve the quadratic equation using any convenient method. $$ 2 x^{2}+7=2 x^{2}-x-4 $$

Step-by-Step Solution

Answer

The solution to the equation \(2 x^{2}+7=2 x^{2}-x-4\) is \(x = 3\)

Step 1: Simplify The Equation

First, simplify the equation by removing the common terms from both sides and moving the remaining terms to one side to set the equation to zero. The given equation is \(2 x^{2}+7=2 x^{2}-x-4\). Subtract \(2x^{2}\) from both sides of the equation: This gives \(7=-x-4\)

Step 2: Rearrange The Equation

Next, the equation needs to be rearranged in the form of a standard quadratic equation \(ax^{2}+bx+c=0\). To achieve this, subtract 7 from both sides and multiply the equation by -1. So we get \(x+4-7=0\), which simplifies to \(x - 3 = 0\)

Step 3: Solve For X

Finally, solve for x. Here as there is no quadratic term, we can directly solve for x by adding 3 to both sides, which gives \(x = 3\). In this case, the equation is linear, so there is only one solution.

Key Concepts

Simplification

Simplification is an essential first step in solving equations, including quadratic ones. It involves making the equation easier to work with by reducing it to its simplest form. To simplify the given equation, we look for terms that can be combined or removed. In our exercise, the original equation is \[2x^2 + 7 = 2x^2 - x - 4\].

Notice the \(2x^2\) term appears on both sides. By subtracting \(2x^2\) from each side, these terms cancel out, which reduces complexity. What remains is \[7 = -x - 4\].

Simplification helps reveal the core equation by eliminating terms that do not affect its fundamental structure, making it easier to solve the problem at hand. In this way, we can often eliminate extraneous or redundant components that may cloud the process.

Standard Form

Putting an equation into its standard form is crucial for clearly identifying its components. For quadratic equations, this form is \[ax^2 + bx + c = 0\].

Once simplified, the equation we dealt with became \[7 = -x - 4\].

To rearrange it into a standard format, you need to collect all terms on one side, resulting in \[0 = -x - 4 - 7\]. Combining like terms, we rearrange this to \[x + 4 - 7 = 0\], which further simplifies to \[x - 3 = 0\].

Though our original equation turns out to be linear after simplification, the process of converting to standard form remains the same for more complicated quadratic terms. Identifying and following this structure is vital for applying formulas or techniques, such as the quadratic formula or factoring, to find solutions.

Linear Equation

A linear equation is an equation that makes a straight line when it's graphed and typically features a variable to the first power. When the terms involving \(x^2\) were eliminated in the simplification step, our equation became linear rather than quadratic.

The equation \[x - 3 = 0\] represents a simple linear equation, showcasing only one variable term and a constant. To solve, you only need basic algebraic manipulation: simply add 3 to each side to isolate \(x\), resulting in \[x = 3\].

Linear equations are straightforward because they have at most one solution, compared to quadratic equations, which can have two solutions. Recognizing that an equation is linear allows quick and efficient problem-solving, reducing the chances for error and confusion.