Problem 97 - Algebra and Trigonometry

Question

Solve equation 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: Rewrite the equation into standard form

A quadratic equation is typically written as \(ax^{2}+bx+c=0\), where a, b, and c are constants. In this case, the equation given is \(4x^{2}-16=0\). We can rewrite it as \(4x^{2}-16+16=0+16\), simplifying to \(4x^{2}=16\).

Step 2: Divide by the coefficient of \(x^{2}\)

Now, divide the entire equation by the coefficient of \(x^{2}\) which is 4. So, \(x^{2}= \frac{16}{4}\), simplifying to \(x^{2}=4\).

Step 3: Apply the square root property

Applying the square root property to finish solving for \(x\), we get \(x = \pm\sqrt{4}\).

Step 4: Solve for \(x\)

This simplifies to \(x = \pm2\)

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations. These equations are anything in the form of \( ax^2 + bx + c = 0 \). The quadratic formula is expressed as: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula allows us to find the solutions for any quadratic equation. It works by discovering the roots of the equation, which are the values of \( x \) where the equation equals zero.

Using the quadratic formula is especially useful when factoring is difficult or impossible. To use this formula effectively:

• Identify the coefficients: \( a \), \( b \), and \( c \).
• Calculate the discriminant: \( b^2 - 4ac \).
• Plug these values into the formula to solve for \( x \).

In the original exercise, using the quadratic formula is not necessary due to its simple nature. Instead, simpler methods were adopted for efficiency.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is crucial for many solution methods, including using the quadratic formula. It is written as \( ax^2 + bx + c = 0 \). Having the equation in this form makes it easier to identify the coefficients needed for solving. For instance, in the equation from the exercise \( 4x^2 - 16 = 0 \), it doesn't initially appear in the standard form due to the missing \( bx \) term.

However, it can be viewed as \( 4x^2 + 0x - 16 = 0 \), where \( a = 4 \), \( b = 0 \), and \( c = -16 \), highlighting the role of each coefficient.

When an equation is not in standard form, it can be restructured. This involves moving all terms to one side of the equation, ensuring the other side equals zero. Properly arranging the equation allows you to clearly see its structure, making methods like factoring or completing the square more accessible.

Square Root Property

The square root property is a convenient method for solving simple quadratic equations, especially those that can be directly isolated as \( x^2 = k \). This property states that if \( x^2 = k \), then \( x = \pm\sqrt{k} \). It's used when the equation lacks a linear \( bx \) term, allowing for straightforward solutions.

In the exercise, once the equation is simplified to \( x^2 = 4 \), the square root property is perfect for the solution. It leads directly to \( x = \pm2 \). This works because:

• The square root of 4 is 2.
• Moreover, \( x \) can be both positive and negative 2, as both are solutions that satisfy the original quadratic equation.

The square root property is a quick and efficient method when applicable, providing real solutions in cases where quadratic equations are structured favorably.