Problem 5 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems 1 through 7, give exact answers, not numerical approximations. Solve: \(x^{2}+1=6\)

Step-by-Step Solution

Answer

The solution to the equation \(x^{2} + 1 = 6\) is \(x = \pm \sqrt{5}\).

Step 1: Simplify the Equation

The first step is to rewrite the equation in a simpler form. This can be done by subtracting 1 from both sides of the equation. The result would be \(x^{2} = 6 - 1\).

Step 2: Calculate

The next step is to calculate the right side of the equation. So, \(x^{2} = 5\).

Step 3: Take the square root

To isolate 'x', take the square root of both sides of equation. Remember, there would be two solutions as square root of a number can be both positive and negative. This gives \(\sqrt{x^{2}} = \sqrt{5}\) which results in \(x = \pm \sqrt{5}\).

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions that relate a squared variable to constants and possibly a linear term. They are of the form \(ax^2 + bx + c = 0\) where \(a\), \(b\), and \(c\) are constants and \(^2\) indicates that the variable \(x\) is squared. To solve these equations, we often rearrange them into a more workable form, setting the equation equal to zero. This allows the use of methods like factoring, completing the square, using the quadratic formula, or—as in the original exercise—by isolating the square term.

Solving a quadratic can sometimes be simple, especially when it lacks a linear \(x\) term and does not have a constant. In the example \(x^2 + 1 = 6\), the process begins by setting \(x^2\) equal to a constant, making the rest of the process of finding \(x\) straightforward. By understanding that all quadratic equations conform to this general format, one can better understand the relationship between the variable and the constants involved which guides the approach for finding the variable's value.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. Represented by the symbol \(\sqrt{}\), square roots are important in solving equations where the variable is squared, like quadratic equations. If we have \(x^2 = n\), then \(x\) is either \(\sqrt{n}\) or \(-\sqrt{n}\). It's crucial to remember that squaring a negative number results in a positive number, so both a positive and a negative root are considered for the variable 'x'.

The concept of square roots is further utilized when the quadratic equation doesn't have a 'perfect square'—a number which has an integer as its square root, like 4, 9, or 16. For such instances, as in \(x^2 = 5\), we retain the square root form \(\pm \sqrt{5}\) to indicate both exact roots. Recognizing this is crucial for understanding that square roots can result in irrational numbers, which cannot be represented as a simple fraction.

Exact Solutions

When solving equations, an 'exact solution' is the unaltered root of the equation, often displayed in radical or fractional form rather than a decimal approximation. Providing exact solutions is particularly important in mathematics because it maintains the precision of the value. This is exemplified by the \(x^2 = 5\) example, where the solution \(x = \pm \sqrt{5}\) is given instead of an approximate decimal like ±2.236.

Exact solutions are valuable in contexts where precision is paramount, like in engineering calculations or theoretical proofs. When a problem states that an 'exact answer' is required, as in our textbook exercise, one must resist the urge to approximate. The solution remains in its radical form or as a fraction to ensure that no detail is lost during computation. Providing the answer in the form of \(\pm \sqrt{5}\) instead of an approximate decimal maintains the integrity and exactitude of the solution.