Problem 26 - Precalculus Student Solutions Manual 5th

Question

Solve each equation by the square root property. $$x^{2}=121$$

Step-by-Step Solution

Answer

x = 11 and x = -11

Step 1: Understand the Square Root Property

The square root property states that if \[ x^2 = k \], then \[ x = \pm \sqrt{k} \]. It means that any number squared that equals a given number has two solutions: the positive and negative square roots of the given number.

Step 2: Apply the Square Root Property

Given the equation \[ x^2 = 121 \], we apply the square root property to find \[ x = \pm \sqrt{121} \].

Step 3: Calculate the Square Root

Find the square root of 121. \[ \sqrt{121} = 11 \]. Therefore, \[ x = \pm 11 \].

Step 4: Write the Solution

The solutions to the equation \[ x^2 = 121 \] are \[ x = 11 \] and \[ x = -11 \].

Key Concepts

solving quadratic equations

Quadratic equations are special types of polynomial equations, and they come in the form of \[ ax^2 + bx + c = 0 \]. In this exercise, we focus on a simple case where the equation takes the form \[ x^2 = k \]. This forms the basis of solving more complex quadratics.
To solve quadratics in this form, we use the square root property. This property helps us find the roots, or solutions, for \[ x \] when \[ x^2 \] equals a number. Remember, equations like these often have two solutions because both positive and negative numbers can square to the same result.
By mastering simpler forms of quadratic equations, you'll build a strong foundation for tackling more advanced problems in precalculus and beyond.

square roots

Square roots are numbers that produce a specified number when multiplied by themselves. The square root of \[ 121 \] is \[ 11 \] because \[ 11 \times 11 = 121 \]. Equations in the form \[ x^2 = k \] can be solved by taking the square root of both sides.
When applying the square root property, we must remember that every positive real number has two square roots: one positive and one negative. This is why applying the square root to \[ x^2 = 121 \] gives us \[ x = \pm 11 \].
Understanding square roots is essential when dealing with quadratic equations, enabling you to break them down easily.

precalculus

Precalculus is designed to prepare you for calculus, and gaining proficiency in concepts like quadratic equations and square roots is vital.
Precalculus will often involve solving quadratic equations, mastering the use of the square root property, and familiarizing yourself with functions and their properties.
The exercise of solving \[ x^2 = 121 \] using the square root property is a fundamental skill. It not only strengthens your algebraic skills but also prepares you for more complex problems that involve quadratic functions and their various properties in delving deeper into calculus.