Problem 24 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation by extracting square roots. List both the exact answer and a decimal answer that has been rounded to two decimal places. $$ x^{2}=144 $$

Step-by-Step Solution

Answer

The solutions to the equation \(x^{2}=144 \) are \(x = 12 \) and \(x = -12 \).

Step 1: Interpret the Equation

This is a quadratic equation given in the form \(x^{2} = 144\). The objective is to find the value or values of \(x\) that satisfy this equation.

Step 2: Find the Square Root

To find the value of \(x\), we need to take the square root of both sides of the equation. In mathematics, the square root of a number has both a positive and negative value. Hence, \(x = \sqrt{144}\) gives \(x = 12 \) and \(x = -\sqrt{144} \) gives \(x = -12 \).

Step 3: Decimal Approximation

Both the solutions \(x=12\) and \(x=-12\) are already in decimal form and do not require any rounding off.

Key Concepts

Extracting Square Roots

When solving quadratic equations by extracting square roots, it involves identifying a perfect square on one side of the equation. In our example, the equation is given as \(x^2 = 144\). With an equation of this form, your goal is to get \(x\) by itself.

To do this, you take the square root of both sides of the equation. It's key to remember that squaring a number always gives a positive result. But when you take the square root of a number, you need to consider both the positive and negative roots. Make sure to express the solutions correctly:

• The positive root: \(x = \sqrt{144} = 12\)
• The negative root: \(x = -\sqrt{144} = -12\)

These are the two possible values of \(x\) that solve the equation. This method only works directly when the quadratic equation is in the form where one side is a perfect square.

Exact and Decimal Answers

Finding exact and decimal answers is an essential skill in mathematics, particularly when dealing with quadratic equations. With the equation \(x^2 = 144\), we find that the exact values for \(x\) are 12 and -12. "Exact answer" means using the precise values obtained without any alterations. Here, both \(x = 12\) and \(x = -12\) are already precise as they are integers.

When equations result in non-integer values, we use decimal forms to represent them in a familiar format. Decimal answers often involve rounding off the results to a specified number of decimal places for simplicity and ease of understanding. Fortunately, 12 and -12 are already in their simplest decimal forms, so no further conversion or rounding is necessary here.

However, in other cases where non-perfect squares are involved, using a calculator assists in converting the square root into a decimal approximation.

Square Root Properties

Understanding the properties of square roots is crucial when dealing with quadratic equations. These properties are what allow us to solve these equations by extracting roots effectively.

• A square root function involves finding a number \(y\) that, when multiplied by itself, gives the original number \(x\). For example, \(\sqrt{144} = 12\) because \(12 \times 12 = 144\).
• Square roots of positive numbers always have two results: one positive and one negative. So, for the solution to \(x^2 = 144\), the roots are \(x = 12\) and \(x = -12\).

Additionally, it's important to note that square root can only be directly applied to non-negative numbers. The understanding of these properties not only aids in solving equations but also helps in recognizing valid and invalid moves in algebraic manipulations.