Problem 23 - Algebra 1

Question

Solve the equation algebraically. Check the solutions graphically. $$ 4 x^{2}=100 $$

Step-by-Step Solution

Answer

The solutions to the equation \(4x^2 = 100\) are \(x = 5\) and \(x = -5\).

Step 1: Isolate x^2

Firstly, isolate \(x^2\) by dividing both sides of the equation by 4. This will result in \(x^2 = 25\).

Step 2: Obtain the roots

To isolate \(x\), take the square root of both sides. Remember there are two solutions for \(x\) because \(x^2=25\) is the same as \(x = \pm\sqrt{25}\). Thus, \(x = \pm5\).

Step 3: Check graphically

Plot the equation \(4x^2 = 100\), and observe that the points at which the graph intersects the x-axis. If the points of intersections occur at x = 5 and x = -5, this confirms the solutions obtained algebraically.

Key Concepts

Algebraic Methods

When solving quadratic equations like 4x^2 = 100, algebraic methods give a clear step-by-step procedure to find the solution. The main goal is to isolate the variable and make x the subject of the formula. The first step is to simplify the equation which, in this case, means dividing both sides by 4 to find x^2 = 25.

Next, we look for the square root of both sides because the variable is squared. This is a critical step as it will yield two solutions, positive and negative: x = \( \pm\sqrt{25} \), which simplifies to x = \pm5. It's important to include both solutions, as quadratic equations inherently have two roots due to their parabolic nature. These algebraic steps ensure that students can handle similar problems where the square of a variable is equal to a number.

Graphical Methods

Graphical methods are a visually intuitive approach to understanding the solutions of a quadratic equation. By plotting the quadratic function 4x^2 = 100 on a graph, one can easily see where the curve crosses the x-axis. These points of intersection represent the solutions to the equation.

Using a graphing calculator or software, plot the equation and check where it hits the x-axis. In this example, the graph will intersect the axis at x = 5 and x = -5, which corresponds with the algebraic solutions previously found. This graphical confirmation is a great cross-verification tool for students to ensure their algebraic solutions are correct.

Square Roots

Understanding square roots is essential when solving quadratic equations like x^2 = 25. To solve for x, we must take the square root of both sides of the equation. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 * 5 = 25.

Keep in mind that squaring a number always gives a positive result, so the square root could be both positive and negative, leading to two solutions, x = 5 and x = -5. Students should be careful to consider both 'roots' because neglecting the negative root can result in incomplete solutions.

Isolate Variable

To isolate a variable means to rearrange an equation so that the variable we're solving for stands alone on one side of the equal sign while keeping the equation balanced. In our example, to isolate x, we must eliminate the coefficient that stands before x^2, which is done by dividing the entire equation by 4. This simplifies down to x^2 = 25.

Once the variable is squared as x^2, we use square roots to further isolate x by itself. Students should understand that 'isolating the variable' is a fundamental concept in algebra that allows you to find the value of the unknown quantity in an equation. It's one of the first steps towards solving complex mathematical problems.