Problem 9 - Algebra 1

Question

Solve the equation algebraically. Check the solution graphically. $$ 3 x^{2}=27 $$

Step-by-Step Solution

Answer

The solutions of the equation are \( x = 3 \) and \( x = -3 \)

Step 1: Simplify Equation

Divide both sides of the equation by 3 to isolate \( x^{2} \) on one side of the equation. That gives \[ x^{2} = 9 \]

Step 2: Solve for x

To solve for \( x \), take the square root of both sides. Recall that when taking the square root of both sides in an equation, you obtain both a positive and a negative solution. Therefore, \( x = 3 \) and \( x = -3 \).

Step 3: Check Solution Graphically

Plot the equation \( y = 3x^{2} \) and a horizontal line at \( y = 27 \). The intersection points should match the calculated solution of \( x = 3 \) and \( x = -3 \).

Key Concepts

Graphical Solution

Graphical solutions provide a visual method to solve equations. By plotting graphs, we can easily see where lines intersect to find solutions. In this problem, we deal with the quadratic equation \( 3x^2 = 27 \). To solve it graphically, we transform the equation into the function \( y = 3x^2 \) and set it equal to \( y = 27 \).

• First, draw the curve for \( y = 3x^2 \). It is a parabola opening upwards.
• Then, plot the horizontal line \( y = 27 \), which is a straight line parallel to the x-axis.

The points where these graphs intersect show the solutions to the equation. By locating the intersection points on the x-axis, we confirm that the solutions are \( x = 3 \) and \( x = -3 \). Graphical representation is useful because it visually confirms the algebraic solution and provides an intuitive understanding of equations.

Square Root Method

The square root method is one of the simpler approaches to solving quadratic equations, especially when they are in the form \( x^2 = k \). Here, our equation from step 1 becomes \( x^2 = 9 \) after simplification.

To use the square root method:

• Take the square root of both sides of the equation: \( \sqrt{x^2} = \sqrt{9} \).
• Remember that the square root operation yields two values: a positive and a negative root.

This results in \( x = 3 \) and \( x = -3 \).

Using the square root method is effective for equations that can be easily reduced to the squared form, providing quick and straightforward solutions without the need for more complex operations or graphing.

Equation Solving

Solving equations involves finding values for unknown variables that make the equation true. For quadratic equations, like \( 3x^2 = 27 \), we have a specific process. First, simplify the equation. Divide by 3, resulting in \( x^2 = 9 \). From here, solving involves methods like factoring, completing the square, or using the quadratic formula, but the square root method is simplest in this case.

The steps are:

• Transform the equation to a simpler form. Ensure all terms involving \( x \) are on one side.
• Use mathematical operations to isolate the variable, applying the square root method since it's suited for this problem.

Finally, we validate our solution both algebraically and graphically. Checking our solutions graphically by plotting the equation helps to ensure accuracy and provides a visual verification.