Problem 18 - Basic Technical Mathematics with Calculus

Question

Use a calculator to solve the given equations. Round solutions to the nearest hundredth. If there are no real roots, state this. $$5-x^{2}=0$$

Step-by-Step Solution

Answer

The solutions are approximately \(x = 2.24\) and \(x = -2.24\).

Step 1: Understanding the Equation

The equation given is a simple quadratic equation in the form of \[ 5 - x^2 = 0 \] This can be rearranged to find when the square of \(x\) equals 5.

Step 2: Reorganizing the Equation

Rearrange the equation so that the unknown is isolated on one side. We want to find \[ x^2 = 5 \] This suggests that \(x\) could be either positive or negative, as squaring both will yield 5.

Step 3: Solving for x

Take the square root of both sides to solve for \(x\): \[ x = \pm \sqrt{5} \] These are the two potential solutions in terms of square roots.

Step 4: Calculating Using a Calculator

Use a calculator to find the approximate values of \(x = \pm \sqrt{5}\). Calculate:- \(\sqrt{5} \approx 2.236\) rounded to the nearest hundredth becomes \(2.24\).- Thus, \(x = 2.24\) or \(x = -2.24\).

Key Concepts

Solving Quadratic Equations

Quadratic equations form an essential part of algebra and are equations of the form \( ax^2 + bx + c = 0 \). In our exercise, the specific equation is \( 5 - x^2 = 0 \). This is a simple type because it lacks the linear \( x \) term and the constant \( c \) is already isolated. To solve it, we rearrange it to find \( x^2 = 5 \), setting the equation in a clear square root format. Solving it involves simple steps and highlights one of the beauties of quadratic equations: they often have two solutions. This happens because squaring a number makes it positive, whether the original number was positive or negative. Here, the solutions are \( x = \sqrt{5} \) and \( x = -\sqrt{5} \). Consider these as the values of \( x \) that make the equation true.

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. For example, \( \sqrt{5} \) means the number that, when multiplied by itself, totals 5. Calculating the square root by hand can be challenging, so calculators are often used for precision.
Square roots can be both positive and negative because both \( 2.236 \cdot 2.236 = 5 \) and \( -2.236 \cdot -2.236 = 5 \). Thus, taking the square root in an equation context, like quadratic equations, generally leads to two answers. It is also crucial to remember that not all numbers have real square roots. For instance, the square root of a negative number results in an imaginary number, which is outside real number solutions. In our exercise, since \( x^2 = 5 \) is positive, real solutions exist.

• Always remember that \( \sqrt{} \) gives two answers: positive and negative.
• For real solutions, the number under the root, or radicand, must be non-negative.

Rounding Numbers

Rounding numbers involves adjusting them for easier understanding and usage, especially when dealing with decimals. The rule followed is simple: look at the digit immediately after the desired place value. If this digit is 5 or more, round up the target digit. For instance, to round \( \sqrt{5} \ approximately \ 2.236 \) to the nearest hundredth, the result becomes 2.24 because the third decimal place digit, 6, is more than 5.
Rounding solutions in math ensures clarity and simplicity, yet always maintain accuracy to the necessary place value as specified in your problem. This practice is vital in both academic settings and real-life applications to facilitate calculations and communication.

• Round the figure based on the value of the immediate next decimal.
• In calculations, state the precision needed (like the nearest hundredth, tenth, etc.).

It's also helpful to maintain consistency in rounding across your calculations for uniformity and error reduction.