Problem 22 - 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. $$x^{2}=144$$

Step-by-Step Solution

Answer

The exact solution of the equation is \(x = 12\) and \(x = -12\), and the decimal solutions rounded to the nearest hundredth are \(x = 12.00\) and \(x = -12.00\).

Step 1: Setup Equation

We start with the equation \(x^{2}=144\).

Step 2: Extracting Square Roots

Since we have a squared term on the left side, we can obtain the value of \(x\) by taking the square root of both sides of the equation. Remember, when taking the square root of the square you obtain two results, a positive and a negative root. Therefore, \(x= \pm \sqrt{144}\).

Step 3: Calculate Exact Solutions

Now, we compute the exact solutions. The Square root of 144 is 12. Hence, we have \(x = \pm 12\). That is, \(x = 12\) or \(x = -12\). Those are the exact solutions.

Step 4: Compute Decimal Solutions

The decimal solutions rounded to the nearest hundredth for \(x = 12\) and \(x = -12\) remain \(x = 12.00\) and \(x = -12.00\), respectively. These are decimal solutions.

Key Concepts

Square Roots

Square roots are fundamental to solving equations like quadratic equations, where we seek to find values that, when squared, yield a given number. For instance, in the problem where \(x^2 = 144\), the operation to find \(x\) involves taking the square root of 144.

• The square root of a number \(a\) is a value \(b\) such that \(b^2 = a\).
• In simple terms, the square root operation "reverses" the squaring of a number.
• For positive numbers like 144, the square root will always produce a value that, when squared, gives the original number.

Understanding how and when to utilize square roots is crucial for solving quadratic equations by extraction of roots, as it simplifies the process of letting us find potential solutions directly.

Exact Solutions

An exact solution in mathematics is the precise answer to a problem without involving approximations or rounding.In our context, when we solve \(x^2 = 144\), we determine that the square root of 144 is 12. Consequently, the exact solutions are \(x = 12\) and \(x = -12\).

• Exact solutions provide the true roots of the equation.
• These roots are important in both mathematical proofs and in practical applications where precision is paramount.

Always note that exact solutions can include values that are positive, negative, or sometimes complex depending on the nature of the original equation.

Decimal Solutions

Decimal solutions involve representing numbers in decimal form, which is especially useful when the exact solution is not a whole number, or when clarity in presentation is necessary. In the exercise where \(x^2 = 144\), the exact roots \(x = 12\) and \(x = -12\) are already whole numbers. However, to express these in decimal form:

• We write them as \(x = 12.00\) and \(x = -12.00\).

This approach makes it easier in scenarios requiring a uniform numerical format, such as in scientific or financial calculations.Decimal solutions must be careful about rounding; especially when the numbers are not whole, rounding is often employed to fit the specific requirements of accuracy, such as to the nearest hundredth or another decimal place.

Positive and Negative Roots

In quadratic equations like \(x^2 = 144\), the roots of the equation are not just positive. They can be a pair, consisting of both positive and negative values. This characteristic stems from the squares of both positive and negative numbers equaling the same positive number.

• For \(x^2 = 144\), both 12 and -12 are valid solutions since \((12)^2 = 144\) and \((-12)^2 = 144\).
• This understanding is essential because it reminds us that every positive whole number, when squared, has both these roots.

Being aware of the instance of positive and negative roots ensures completeness in solutions, especially in theoretical mathematics where missing a root could lead to incorrect conclusions. This occurrence is a direct result of the equation's symmetry regarding positive and negative values.