Problem 97 - College Algebra Essentials

Question

Solve each equation in Exercises \(83-108\) by the method of your choice. $$ 4 x^{2}-16=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 2\) and \(x = -2\).

Step 1: Rearrange the equation

Rearrange the equation to the form \(x^{2} = m\). We first add 16 to both sides of the equation, which gives us \(4x^{2} = 16\). We can then divide by 4 on both sides to get \(x^{2} = 4\).

Step 2: Solve for \(x\)

Now, take the square root on both sides. This gives us two solutions: \(x = +\sqrt{4}\) and \(x = -\sqrt{4}\). Remember that when \(x^{2} = m\), \(x\) can be both \(+\sqrt{m}\) and \(-\sqrt{m}\). So in this case, we get \(x = 2\) and \(x = -2\).

Key Concepts

Solving Quadratic Equations

Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. Solving these equations involves finding the values of \( x \) that make the statement true.
For the given exercise, we have \( 4x^2 - 16 = 0 \). Our goal is to manipulate this equation until we can find the values of \( x \). Quadratic equations can often have two solutions, reflecting the parabolic nature of their graphs.
Understanding the structure of quadratic equations is key to effectively solving them using various methods. There are multiple techniques, like factoring, using the quadratic formula, and completing the square, in addition to the square root method discussed in this exercise.

Square Root Method

The square root method is a straightforward technique for solving quadratic equations that can be rearranged into the form \( x^2 = m \).
Here's how it works:

• First, adjust the equation so that \( x^2 \) is isolated on one side.
• Then, take the square root of both sides of the equation.
• Always remember to consider both the positive and negative roots as solutions, since squaring either a positive or a negative number results in a positive number.

In the example \( 4x^2 - 16 = 0 \), we rearranged it to \( x^2 = 4 \). By taking the square root, we found \( x = +2 \) and \( x = -2 \), representing the two solutions. The square root method is particularly useful when the quadratic equation is easily rearranged to feature a perfect square.

Algebraic Manipulation

Algebraic manipulation is a fundamental skill in solving equations, particularly for rearranging terms to find a solution.For quadratic equations,

• We often need to add or subtract terms from both sides to isolate our variable.
• We might multiply or divide both sides by a constant to simplify the equation.

In our exercise, the initial equation was \( 4x^2 - 16 = 0 \). By adding 16 to both sides, we get \( 4x^2 = 16 \). Next, dividing both sides by 4 simplifies it to \( x^2 = 4 \). The key to effective algebraic manipulation is maintaining balance, ensuring whatever operation is performed on one side is also done to the other.
This process of methodically adjusting the equation helps us get closer to the solution by simplifying complex relationships into more manageable forms.