Problem 30 - Algebra 1

Question

Solve the equation algebraically. Check the solutions graphically. $$ x^{2}-53=11 $$

Step-by-Step Solution

Answer

The solutions to the equation \(x^{2} - 53 = 11\) are x = +8 and x = -8.

Step 1: Simplify the given equation

We have the equation \(x^{2} - 53 = 11\). We will first simplify this equation by adding 53 to both sides of the equation. This will leave us with the equation \(x^{2} = 64\).

Step 2: Solve for x

The next step is to solve for x by taking the square root of both sides. Remember that the square root of a number yields two possible solutions: one positive and one negative. In this context, we get \(x = +8\) and \(x = -8\). Therefore x can be either of these two.

Step 3: Confirm the solution graphically.

The graph of the function \(f(x) = x^{2} - 53\) is a parabola. If the given solutions are correct, then the points (8, 11) and (-8, 11) should be on the parabola. You can confirm this by plotting the equation on graph paper or using a graphical calculator. In this case, we find that both points indeed fall on the line of the graph, therefore confirming that the solutions are correct.

Key Concepts

Graphing Parabolas

A parabola is a U-shaped curve that can open upwards or downwards on a graph. When dealing with quadratic equations, the graph of the equation is typically a parabola. For the function \( f(x) = x^2 - 53 \), the graph will open upwards because the coefficient of \( x^2 \) is positive.
To graph a parabola, it is useful to find important points such as the vertex, and the x-intercepts (or roots). In our example, after solving the algebraic equation, we know the roots are \( x = 8 \) and \( x = -8 \). These roots indicate where the parabola crosses the x-axis.
Key features of a parabola you should know:

• The vertex is the highest or lowest point on the graph. In this case, it's a minimum point because the parabola opens upwards.
• The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Here, it is the line \( x = 0 \).
• The direction of the parabola (upwards or downwards), which depends on the sign of the leading coefficient.

Visualizing the parabola will help you confirm solutions to quadratic equations and understand the behavior of these functions.

Square Root Method

The square root method is a technique used to solve quadratic equations that can be simplified to the form \( x^2 = c \). This is seen in our example, where the equation simplifies to \( x^2 = 64 \). The goal of this method is to isolate \( x \) by taking the square root of both sides of the equation.
It's crucial to remember:

• Taking the square root of a number results in two possible solutions: one positive and one negative. Hence, for \( x^2 = 64 \), the solutions are \( x = 8 \) and \( x = -8 \).
• This method is best applied when the quadratic equation can be easily laid out as \( x^2 = c \). Otherwise, you might need to rearrange the equation first.

Using the square root method is often simple and straightforward, especially when the quadratic equation is already in a convenient form or can be easily modified to become one. It allows you to bypass more complicated algebraic manipulations.

Algebraic Solutions

Algebraic solutions involve manipulating the equation using algebraic rules to find the value(s) of \( x \). For the given equation \( x^2 - 53 = 11 \), the process begins with simplifying the equation. Adding 53 to both sides, we get \( x^2 = 64 \). This step is critical in paving the way for the next phase: solving by the square root method.
Why are algebraic solutions useful?

• They provide an exact numerical answer, unlike some methods that rely on approximations.
• They lay the groundwork for verifying solutions graphically, offering a chance to see the equation from both analytical and visual perspectives.
• Learning algebraic techniques helps develop problem-solving skills and enhances understanding of mathematical functions.

By mastering algebraic solutions, you increase your capability to solve a wide variety of equations, building a solid foundation in algebraic reasoning.