Problem 16 - A Graphical Approach to College Algebra

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$2 x^{2}=48$$

Step-by-Step Solution

Answer

The solutions are \( x = 2\sqrt{6} \) and \( x = -2\sqrt{6} \).

Step 1: Rearrange the Equation

Start by dividing both sides of the equation by 2 to simplify it. So, the equation becomes:\[ x^2 = 24 \]

Step 2: Solve for x

To solve for \( x \), take the square root of both sides. Remember when you take the square root, you consider both the positive and negative solutions:\[ x = \pm\sqrt{24} \]

Step 3: Simplify the Square Root

Simplify \( \sqrt{24} \) by factoring it into \( \sqrt{4 \cdot 6} = \sqrt{4} \times \sqrt{6} \), which simplifies to:\[ x = \pm 2\sqrt{6} \]

Step 4: Verify Solutions Graphically

To verify graphically, plot the equation \( y = 2x^2 \) and \( y = 48 \) on a graph. The x-coordinates where these graphs intersect provide the solutions to the equation. The root \( x = \pm 2\sqrt{6} \) should correspond to the intersection points of the graphs.

Key Concepts

Solving Quadratic Equations

Quadratic equations are equations involving the squared term of the variable, in this case, denoted as \( x^2 \). They are generally of the form \( ax^2 + bx + c = 0 \). In our given problem, the equation is simpler: \( 2x^2 = 48 \). To solve a quadratic equation like this, you need to isolate the variable \( x \).

The simplest way to do this is to first divide through by the coefficient of \( x^2 \), making it simpler to manage. Here, dividing by 2, we transform the equation to \( x^2 = 24 \).

Once we have \( x^2 = 24 \), the task is to solve for \( x \) by taking the square root of both sides. Always remember when taking square roots in algebra that you will encounter two possible solutions: one positive and one negative. Thus, the solutions are \( x = \pm \sqrt{24} \). These values of \( x \) solve the quadratic equation.

Graphical Solutions

Graphical solutions involve plotting the equation on a coordinate system to visually determine the solutions. For the equation \( 2x^2 = 48 \), represent it graphically by plotting \( y = 2x^2 \).

Similarly, plot \( y = 48 \) as a horizontal line across the graph. The intersections of these plots with the x-axis indicate where \( x \) satisfies \( 2x^2 = 48 \). In our scenario, the x-coordinates at these intersections give the solutions \( x = \pm 2\sqrt{6} \).

• The graph of \( y = 2x^2 \) is a parabola opening upwards.
• The graph of \( y = 48 \) is a straight horizontal line.
Such graphical methods offer intuitive, visual validation of the solutions you find analytically. They also illustrate where the function \( y = 2x^2 \) reaches certain values.

Square Roots

Square roots are an operation applied to numbers to yield a value which, when squared, returns the original number. In solving quadratic equations, taking the square root of both sides is often a step to isolate \( x \).

In the equation \( x^2 = 24 \), applying square roots gives \( x = \pm\sqrt{24} \). Simplifying square roots can often break them down into more manageable terms. In this case:

\( \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} \). As \( \sqrt{4} = 2 \), we obtain \( 2\sqrt{6} \).

• Square roots can simplify expressions involving squares, as seen with \( \sqrt{24} = 2\sqrt{6} \).
• Always consider both positive and negative roots when finding solutions to quadratic equations.
This simplification helps in understanding the precise values for any calculations or graphical solutions.