Problem 53 - Prealgebra

Question

Solve each equation. Round to the nearest tenth, if necessary. $$196=y^{2}$$

Step-by-Step Solution

Answer

The solutions are \(y = 14\) and \(y = -14\).

Step 1: Understand the Equation

The given equation is \(196 = y^2\). We need to solve for \(y\) by finding the value of \(y\) that satisfies the equation.

Step 2: Isolate the Variable

To solve for \(y\), we need to isolate it. Since \(y^2 = 196\), we take the square root of both sides of the equation to get \(y = \pm\sqrt{196}\).

Step 3: Calculate the Square Root

Calculate \(\sqrt{196}\). The square root of 196 is 14, because \(14 \times 14 = 196\).

Step 4: Consider Both Roots

Since \(y = \pm\sqrt{196}\), \(y\) can be either positive or negative. Thus, \(y = 14\) or \(y = -14\).

Key Concepts

Understanding the Square Root

The square root is a fundamental concept in mathematics. It is a value that, when multiplied by itself, gives the original number. For example, the square root of 196 helps us find the number which, when squared, equals 196.

The symbol for square root is \(\sqrt{}\). Therefore, \(\sqrt{196}\) refers to the number that, when multiplied by itself, equals 196. Calculating \(\sqrt{196}\) involves finding 14, because \(14 \times 14 = 196\).

• Square roots can be both positive and negative because squaring either value gives a positive result.
• Hence, \(\sqrt{196}\) could be either 14 or -14.

Understanding how square roots work is crucial in solving equations.

Introduction to Quadratic Equations

A quadratic equation is a second-degree polynomial equation in the form \(ax^2 + bx + c = 0\), where \(a, b,\) and \(c\) are constants and \(a eq 0\). The equation \(y^2 = 196\) is a simplified form of quadratic equations because it involves an unknown raised to the second power.

When solving quadratic equations, you'll often need to find two values of the unknown that satisfy the equation. In the case of \(196 = y^2\), solving requires determining the square root of the constant. This gives us the potential solutions \(y = 14\) and \(y = -14\).

• This method of solving is especially straightforward when the quadratic equation lacks a linear term like our example.
• Understanding the structure and solutions of quadratic equations empowers you to tackle a wide variety of mathematical problems.

Positive and Negative Roots in Equations

When solving equations like \(y^2 = 196\), it is important to remember the principle of positive and negative roots. Since both \(14\) and \(-14\) squared will result in 196, both are considered valid solutions.

In equations involving square roots, the "±" symbol represents that each root has both a positive and negative component.

• This is because squaring any real number, whether positive or negative, results in a positive product.
• Thus, after isolating the variable and computing the square root, always account for both possible values.

Recognizing both roots ensures comprehensive solutions and is key to correctly solving quadratic equations.