Problem 29 - Algebra 2

Question

Solve each equation by using the method of your choice. Find exact solutions. \(4 x^{2}-8=0\)

Step-by-Step Solution

Answer

The solutions are \( x = \sqrt{2} \) and \( x = -\sqrt{2} \).

Step 1: Isolate the Quadratic Term

Start with the original equation: \[ 4x^2 - 8 = 0 \] To isolate the quadratic term, add 8 to both sides of the equation: \[ 4x^2 = 8 \]

Step 2: Solve for x^2

With the quadratic term isolated, divide both sides by 4 to solve for \( x^2 \): \[ x^2 = \frac{8}{4} \]Simplifying the right side gives: \[ x^2 = 2 \]

Step 3: Apply the Square Root

To solve for \( x \), take the square root of both sides.Remember that taking the square root introduces both the positive and negative solutions:\[ x = \pm \sqrt{2} \]

Step 4: Write the General Solution

The exact solutions to the equation are:\[ x = \sqrt{2} \] and \[ x = -\sqrt{2} \].

Key Concepts

Solving Quadratic Equations

Quadratic equations are essential in algebra and appear in various forms. A standard quadratic equation can be written as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. Solving these equations means finding the value(s) for \( x \) that satisfy the equation. Quadratic equations typically have two solutions. These solutions could be real or complex numbers, depending on the coefficients. There are several methods to solve quadratic equations, including:

• Factoring
• Using the quadratic formula
• Completing the square
• Using the square root method

Each method is suitable for different kinds of quadratic equations. Choosing the right technique depends on the specific form the equation takes and the ease of computation.

Square Root Method

The square root method is efficient for solving quadratic equations when they are in the form \( ax^2 = c \), without the linear \( bx \) term. This method involves:

• Isolating the \( x^2 \) term on one side of the equation
• Taking the square root of both sides afterwards

By isolating \( x^2 \), you simplify the equation to \( x^2 = d \), allowing you to apply the square root to find \( x \). This yields two solutions: the positive and negative square roots of \( d \). Remember, every positive number has two square roots: one positive and one negative. So, if \( x^2 = 4 \), \( x \) can be \( 2 \) or \( -2 \). This dual solution is critical in ensuring you capture all potential values satisfying the original quadratic equation.

Step-by-Step Solution

Working through problems in a step-by-step manner can demystify the process, making it easier to understand. Let's apply the method to solve \( 4x^2 - 8 = 0 \): **Step 1**: Begin with isolating the quadratic term. Add \( 8 \) to both sides:\[ 4x^2 = 8 \]**Step 2**: Divide by \( 4 \) to solve for \( x^2 \):\[ x^2 = 2 \]**Step 3**: Take the square root of both sides. This action includes both the positive and negative outcomes:\[ x = \pm \sqrt{2} \]**Final Step**: State the solutions, which are \( x = \sqrt{2} \) and \( x = -\sqrt{2} \). Using clear steps allows students to track their progress and ensure they understand each stage before moving on.

Exact Solutions

Finding exact solutions means obtaining precise values, without approximations. In quadratic equations, exact solutions often involve radicals or integers, not decimals. For the equation \( 4x^2 - 8 = 0 \), we derived the exact answers \( x = \sqrt{2} \) and \( x = -\sqrt{2} \). These are not rounded numbers; they express the solution completely as fractions of integers and square roots. In math, exact solutions are preferred as they maintain the integrity of the original equation and allow for accurate interpretation in subsequent use. When faced with quadratic problems, students should aim for exact solutions unless specified otherwise, as they provide complete and undistorted information.