Problem 15 - A Graphical Approach to College Algebra

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$2 x^{2}=90$$

Step-by-Step Solution

Answer

The solutions are \(x = 3\sqrt{5}\) and \(x = -3\sqrt{5}\).

Step 1: Write the Equation

The given equation is \(2x^2 = 90\). This is a quadratic equation that we need to solve for \(x\).

Step 2: Simplify the Equation

First, divide both sides of the equation by 2 to simplify:\[\frac{2x^2}{2} = \frac{90}{2}\]This reduces to:\[x^2 = 45\]

Step 3: Solve for x

To solve for \(x\), take the square root of both sides of the equation:\[x = \pm \sqrt{45}\]\(\sqrt{45}\) can be simplified to \(3\sqrt{5}\), so:\[x = \pm 3\sqrt{5}\]

Step 4: Graphically Support the Solution

Consider the graph of the equation \(y = 2x^2 - 90\). The solutions occur where this graph intersects the x-axis (i.e., where \(y = 0\)).Plotting \(y = 2x^2 - 90\), you'll notice that the graph intersects the x-axis at \(x = 3\sqrt{5}\) and \(x = -3\sqrt{5}\), confirming the solutions to the equation.

Key Concepts

Graphical Solutions

Graphical solutions offer a visual approach to solving quadratic equations. By plotting the equation on a coordinate plane, you can easily identify where the solutions lie. You do this by determining the points where the graph intersects the x-axis. In the context of the quadratic equation \(2x^2 = 90\), this involves rephrasing it into a function form: \(y = 2x^2 - 90\).

Here's how you can approach it:

• Rearrange the quadratic into a typical format \(y = 2x^2 - 90\), transforming the equation into a function.
• Plot the graph of this function on a coordinate plane.
• The points where the graph crosses the x-axis are your solutions for \(x\).

For this equation, when plotted, you'll see it slopes upwards and meets the x-axis at two points. These intersection points represent the values \(x = \pm 3\sqrt{5}\), aligning with the solutions found algebraically.

This method gives you both a numerical and visual understanding of quadratic equations.

Quadratic Equation Simplification

Simplification is the first step in solving any quadratic equation efficiently. It makes the equation easier to handle mathematically. Let's break down the process of simplifying the given equation \(2x^2 = 90\).

Here's how it works:

• First, aim to isolate the \(x^2\) term. This involves dividing each part of the equation by 2, resulting in \(x^2 = 45\).
• By doing so, you reduce the equation to a standard form, making it manageable for further solving techniques such as the square root method.
• This simplification prepares the equation for solution tactics that effectively reveal the roots of the equation.

These steps are crucial because they transform a complex equation into a more straightforward one, ready for further techniques such as taking square roots to find solutions.

Square Root Method

Employing the square root method is a straightforward way to solve simplified quadratic equations, like \(x^2 = 45\). This method involves taking the square root of both sides of the equation and requires just a few steps:

• First, express the equation so that the variable \(x\) is squared: \(x^2 = 45\).
• Apply the square root to each side, remembering to incorporate both the positive and negative roots. This results in: \(x = \pm \sqrt{45}\).
• Simplify the expression \(\sqrt{45}\) further to obtain \(3\sqrt{5}\). Therefore, you'll arrive at the solutions: \(x = \pm 3\sqrt{5}\).

The square root method is highly effective for equations that are newly simplified to a square term on one side. It offers a quick resolution, especially when paired with simplification steps, and helps to clarify quadratic solutions in a clear, concise manner.