Problem 26 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation by extracting square roots. List both the exact solutions and the decimal solutions rounded to the nearest hundredth. $$x^{2}-3=78$$

Step-by-Step Solution

Answer

The exact solutions are \(x = \pm 9\) and the decimal solutions are \(x = \pm 9.00\).

Step 1: Isolate \(x^{2}\)

Add 3 to both sides to isolate \(x^{2}\). This allows to make mathematical adjustments, transforming equitation into \(x^{2}=81\).

Step 2: Solve for \(x\)

To solve for x, one needs to take the square root of both sides. Remembering that the square root could be positive or negative, this will result in \(x = \pm 9\).

Step 3: Find decimal solutions

As the exact solutions are already integers, no rounding is required. This makes the decimal solutions \(x = \pm 9.00\).

Key Concepts

Quadratic Equations

A quadratic equation is a mathematical expression in the form of \(ax^{2} + bx + c = 0\). It is called 'quadratic' because the highest exponent of the variable \(x\) is two, which is also known as the square. These equations are a key concept in algebra and appear in various forms, solving real-world problems such as calculating areas and physics-related scenarios.
Quadratic equations can have three forms depending on their complexity:

• Standard form: \(ax^{2} + bx + c = 0\)
• Factored form, where the expression is written as a product of its factors
• Vertex form, focusing on the maximum or minimum value of the quadratic function

The equation you worked with, \(x^{2} - 3 = 78\), is already close to the standard form. The goal is to arrange it correctly and isolate terms to solve for \(x\). This typically involves rearranging the terms and using operations like addition and subtraction until only terms with \(x\) are on one side.

Exact Solutions

Exact solutions of quadratic equations are the values of \(x\) that precisely satisfy the equation. These solutions can be integers, fractions, or involve radicals. In our exercise, after isolating \(x^{2}\), we ended with \(x^{2} = 81\). Taking the square root of both sides gives us \(x = \pm 9\).
Exact solutions are significant because they provide a precise answer without approximations. When we find \(x = 9\) and \(x = -9\), we know these are the specific points where the original equation holds true. In calculations, maintaining exact values is crucial for ensuring accuracy in further mathematical operations.

• Exact solutions often help validate decimal results, offering a benchmark for checking accuracy.
• This is more noticeable in complex equations where solutions might take forms like \(\frac{3}{2}\) or \(\sqrt{5}\).

Decimal Solutions

Decimal solutions provide an approximate result, often used for ease of understanding, visualization, or when an application tool requires them. In the solved equation \(x^{2} - 3 = 78\), we derived an exact value of \(x = \pm 9\). Thus, the decimal solutions are \(x = 9.00\) and \(x = -9.00\).
Decimal solutions are convenient in practical scenarios where precise measures are needed. However, one must be cautious when rounding, as inaccuracies can compound in lengthy calculations. The key points about decimal solutions are:

• Useful in scenarios that involve estimation or when exact precision is unnecessary.
• Always consider the degree of rounding needed - in many school problems, this is typically to two decimal places.
• Rounding errors should be minimized by keeping track of more significant figures in between calculations.

When providing decimal answers, ensure to state both the positive and negative solutions as you learned from the exact values.