Problem 22 - Algebra 1

Question

Solve the equation algebraically. Check the solutions graphically. $$ 4 x^{2}=16 $$

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Equation

Firstly, it is necessary to rewrite the given equation \(4x^2=16\) in a simpler form. This can be achieved by dividing the entire equation by 4, which results in the equation \(x^2=4\).

Step 2: Square Root

The next step is to take the square root on both sides of the equation. This will provide the potential solutions for \(x\). Remember that the square root of any number includes both a positive and a negative value, so the solutions should be \(x=2\) and \(x=-2\).

Step 3: Check Solutions Graphically

To verify the obtained solutions graphically, plot the equation \(4x^2=16\). The intersections with the x-axis are the solutions to the equation. The graph should cross the x-axis at \(x=2\) and \(x=-2\), which is in agreement with the algebraic solution.

Key Concepts

Graphical Verification

Graphical verification involves using a graph to confirm the solutions of equations. By plotting the equation, you can visually see where solutions lie. In the given equation, we had to solve for where the graph of \(4x^2 = 16\) intersects the x-axis. Here's how it works:

• Graph the equation \(4x^2 = 16\). This is a parabola opening upwards.
• The x-axis is where \(y=0\). Therefore, intersections indicate solutions where the equation equals zero.
• In our case, the graph intersects the x-axis at \(x=2\) and \(x=-2\). This visually confirms our algebraic solutions.

Visual confirmation is powerful because it not only checks correctness but also enhances understanding by providing a different perspective.

Simplification of Equations

Simplification of equations involves rewriting equations to make them easier to solve. In our exercise, we simplified \(4x^2=16\) to \(x^2=4\). Simplification helps to:

• Identify the core structure of the equation by removing unnecessary terms.
• Make operations like solving, factoring, or graphing more straightforward.

To simplify:

• Look for common factors. Here, we divided through by 4, resulting in \(x^2=4\).
• Check if further simplification is possible or needed.

By simplifying early, solving processes become clearer and less error-prone.

Properties of Square Roots

Understanding the properties of square roots is crucial when solving quadratic equations. When we simplified the equation to \(x^2=4\), the next step required us to take the square root. Here are some important points:

• The square root of a positive number results in two values: one positive and one negative. So, \(\sqrt{4} = 2\) and \(-2\).
• Taking square roots is about finding values which when squared return the original number.
• Square roots help us find roots of the equation. These are the solutions where the parabola intersects the x-axis.

A clear grasp of these properties helps in recognizing that equations like \(x^2=4\) have two solutions, thus ensuring accurate problem-solving.