Problem 10 - Elementary and Intermediate Algebra

Question

For each number, find all of its square roots. $$ 81 $$

Step-by-Step Solution

Answer

The square roots of 81 are 9 and -9.

Step 1 - Understand the Concept of Square Roots

A square root of a number 'a' is a number 'b' such that \(b^{2} = a\). Remember that every positive number has two square roots: one positive and one negative.

Step 2 - Identify the Number

Here, the number given is 81. We need to find both positive and negative square roots of 81.

Step 3 - Find the Positive Square Root

Think of a number that, when multiplied by itself, equals 81. Mathematically, we need to solve for \(b\) in \(b^{2} = 81\). The positive square root of 81 is 9 because \(9^{2} = 81\).

Step 4 - Find the Negative Square Root

Similarly, the negative square root of 81 is -9 because \((-9)^{2} = 81\).

Step 5 - List All Square Roots

Combine your positive and negative results. Therefore, the square roots of 81 are 9 and -9.

Key Concepts

positive square root

A square root is a value that, when multiplied by itself, gives the original number. For instance, the square root of 81 refers to finding a number which, when squared, equals 81. Every positive number has two square roots: one positive and one negative. Here, we'll talk about the positive square root.

The positive square root of a number is the non-negative value 'b' that solves the equation
\( b^2 = a \).

• To find the positive square root of 81, think of a number that, when multiplied by itself, equals 81.
• In this case, 9 is a suitable choice because \(9^2 = 81\).

Thus, the positive square root of 81 is 9.

negative square root

Now, let's discuss the negative square root. Just like the positive square root, the negative square root of a number 'a' is a value 'b' that also satisfies the equation
\( b^2 = a \)
but in this case, 'b' is negative.

• For the number 81, the negative square root is -9, because\((-9)^2 = 81\).
• This means you are looking for a negative number that, when squared, equals 81.

So, the negative square root of 81 is -9.

Remember,

• Both positive and negative square roots are important and should be considered when solving problems involving square roots.

solving quadratic equations

Understanding how to find square roots is essential for solving quadratic equations.
A quadratic equation is typically in the form \(ax^2 + bx + c = 0\). Quadratic equations can be solved by several methods, including factoring, completing the square, and using the quadratic formula.

The quadratic formula \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \) is particularly useful because it applies to all quadratic equations. Here, the square root symbol (\( \sqrt{} \)) comes into play.

• Simplify the expression under the square root (known as the discriminant, \( b^2 - 4ac \)).
• Calculate the positive and negative square roots of the discriminant.
• Substitute these values back into the quadratic formula to find the solutions for 'x'.

By understanding positive and negative square roots, students can more effectively solve quadratic equations and grasp more advanced mathematical concepts.