Problem 25 - Precalculus Student Solutions Manual 5th

Question

Solve each equation by the square root property. $$x^{2}=16$$

Step-by-Step Solution

Answer

x = 4, -4

Step 1 - Understand the Square Root Property

The square root property states that if \(x^2 = k\), then \(x = \pm \sqrt{k}\). This means we need to take the square root of both sides of the equation \(x^2 = 16\).

Step 2 - Apply the Square Root Property

Taking the square root of both sides of \(x^2 = 16\) gives \(x = \pm \sqrt{16}\).

Step 3 - Evaluate the Square Root

Calculate \sqrt{16}. Since \sqrt{16} = 4, the equation becomes \(x = \pm 4\).

Step 4 - Write the Solution

Express the final solution. Thus, the solutions are \(x = 4\) and \(x = -4\).

Key Concepts

solving quadratic equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). They can be solved in several ways, such as factoring, completing the square, using the quadratic formula, or by employing the square root property. For our specific example, we used the square root property. This technique is particularly fast and efficient when the quadratic equation has no linear term (the term with \(bx\) is missing) and can be easily transformed into the form \(x^2 = k\). In such cases, we can isolate the \(x^2\) term and then use the property to solve for \(x\). This approach simplifies the process and quickly leads to the correct solutions.

square roots

A square root is a number which, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because \(4 \times 4 = 16\). The process of finding the square root is denoted by the symbol \(\sqrt{}\). It is important to remember that square roots can be both positive and negative. This is because both \(4 \times 4 = 16\) and \(-4 \times -4 = 16\). Thus, \(\sqrt{16} = 4\) and \(\sqrt{16} = -4\). Always keep in mind this dual nature when solving equations that involve square roots.

positive and negative solutions

When you solve an equation like \(x^2 = 16\) using the square root property, you must include both the positive and negative solutions. This is because squaring either a positive number or a negative number will result in a positive value. So, for \(x^2 = 16\), taking the square root of both sides yields \(x = \pm 4\), giving us two solutions: \(x = 4\) and \(x = -4\). It's essential to write both solutions to fully solve the equation. To summarize, always consider both the positive and negative square roots when dealing with quadratic equations and square roots.