Problem 34 - Prealgebra

Question

Solve each equation. Round to the nearest tenth, if necessary. $$a^{2}=49$$

Step-by-Step Solution

Answer

The solutions are \(a = 7\) and \(a = -7\).

Step 1: Understand the equation

The equation given is \(a^{2} = 49\), which is a quadratic equation since the highest power of the variable \(a\) is 2. Our goal is to find the value of \(a\) that satisfies this equation.

Step 2: Take the square root of both sides

To solve for \(a\), we take the square root of both sides of the equation. This gives us \(\sqrt{a^{2}} = \sqrt{49}\). Remember that taking the square root of both sides introduces both a positive and a negative solution.

Step 3: Simplify the result

Calculating the square root, we find that \(\sqrt{49} = 7\). Therefore, the solutions to the equation are \(a = 7\) and \(a = -7\).

Step 4: Conclude the solution

Since both \(a = 7\) and \(a = -7\) satisfy the equation \(a^{2} = 49\), these are the solutions. As the prompt asks for rounding, if necessary, check if rounding is required. Here, no rounding is needed as the solutions are whole numbers.

Key Concepts

Square Root

The concept of the square root is fundamental when solving quadratic equations like \(a^2 = 49\). The square root of a number \(n\), written as \( \sqrt{n} \), is a value that, when multiplied by itself, gives back \(n\). For instance, \( \sqrt{49} \) is the number that - when multiplied by itself - results in 49.
To solve equations like \(a^2 = 49\), we take the square root of both sides. This operation is denoted by the square root symbol \(\sqrt{}\).

• Taking the square root on both sides: \( \sqrt{a^2} = \sqrt{49} \)
• Simplifying, we find that \( \sqrt{49} = 7 \)

The square root operation is a way to "undo" the squaring of a number. It’s crucial to help us find the values of \(a\) that make \(a^2 = 49\) true. Remember, in mathematics, solutions often come in pairs - the positive and negative roots.

Positive and Negative Solutions

When you take the square root of both sides of a quadratic equation, it's important to consider both positive and negative roots. This is because both \(a = 7\) and \(a = -7\) will satisfy \(a^2 = 49\).
This happens due to the nature of squaring a negative number. For example, (-7) squared is 49, just as (7) squared is also 49.

• \(7 \times 7 = 49\)
• \((-7) \times (-7) = 49\)

Therefore, quadratic equations often have two solutions: one positive and one negative. This dual solution ensures that we capture all possible values of the variable that satisfy the equation.

Whole Numbers

In this equation, \(a^2 = 49\), the solutions have emerged as whole numbers. Whole numbers are numbers without fractions or decimals; they can be both positive and negative, as well as zero.
The solutions \(a = 7\) and \(a = -7\) satisfy this property. Whole numbers make it easier to interpret the answers and often do not require rounding. Hence, in this exercise, there's no need to round the results to the nearest tenth. Solving equations can sometimes yield fractions or decimals, but when you get whole numbers, it means the solutions are precise and straightforward.