Problem 41 - Algebra 1: Concepts and Skills

Question

Solve the equation algebraically. Check your solutions by graphing. $$x^{2}+37=118$$

Step-by-Step Solution

Answer

The solutions to the quadratic equation are \(x = 9\) and \(x = -9\).

Step 1: Arrange the equation

Rearrange the equation to look like a standard quadratic equation \(x^{2}+37=118\), which simplifies to \(x^{2} = 81\) where we subtract 37 from both sides.

Step 2: Square root of RHS

Take the square root on both sides. We get \(x = \pm 9\). Remember, whenever we take square root in algebra, we consider both + and - values.

Step 3: Check by Graphing

To verify your solution(s), graph the equation \(y = x^{2} - 81\). The x-intercepts of the graph should correspond to the solution(s) we found, i.e., \(x = 9, -9\).

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental concept in algebra and are commonly used in various areas of mathematics to model real-world scenarios. These equations take the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). In the equation, \(x\) represents the variable, or the unknown value you need to find.
A core aspect of solving quadratic equations is that they can produce two possible solutions. This is because the graph of a quadratic equation is a parabola, which can intersect the x-axis at two distinct points, one point, or sometimes none at all.

• If the equation intersects the x-axis at two points, it has two real roots.
• If it just touches the x-axis, there is one real double root.
• If it does not touch the x-axis, the roots are complex or imaginary.

To solve a quadratic equation like \(x^2 + 37 = 118\), you first rearrange it to the standard form, resulting in \(x^2 - 81 = 0\). This equation can then be solved using various methods such as factoring, completing the square, or using the quadratic formula.

Graphing Solutions

Graphing is a useful technique to verify the solutions of quadratic equations visually. When we graph a quadratic equation like \(y = x^2 - 81\), the resulting curve is a parabola. The solutions to the quadratic equation are found where this parabola intersects the x-axis, known as the x-intercepts.
For our equation, after rearranging it as \(x^2 - 81 = 0\), graphing \(y = x^2 - 81\) will help us see that there are two x-intercepts at \(x = 9\) and \(x = -9\). This confirms our algebraic solutions:

• When \(x = 9\), \((9)^2 - 81 = 0\)
• When \(x = -9\), \((-9)^2 - 81 = 0\)

Graphing provides a practical way to validate algebraic calculations, ensuring that the solutions make sense. Besides helping us demonstrate the validity of our solutions, graphing also offers insight into the behavior of quadratic functions, such as the width and direction of their parabolas.

Square Roots

The operation of taking square roots is essential when solving quadratic equations, particularly in equations that can be arranged into the form \(x^2 = k\), where \(k\) is a constant. In our example, \(x^2 = 81\), taking the square root of both sides means we seek a number that, when multiplied by itself, results in 81.
Here, the solutions are \(x = 9\) and \(x = -9\) because both \((9)^2 = 81\) and \((-9)^2 = 81\). When taking square roots in algebra, it is important to consider both the positive and negative roots, as both are valid solutions in quadratic contexts.
This duality is rooted in the property of exponents and quadratic equations, where squaring either a positive or negative number results in the same non-negative number. Thus, recognizing that solutions come in pairs is crucial, each with its algebraic significance.