Problem 61 - Algebra 1: Concepts and Skills

Question

Solve the equation. Write the solutions as integers if possible. Otherwise, write them as radical expressions. $$ x^{2}=11 $$

Step-by-Step Solution

Answer

The solutions to the given equation are \(x= \sqrt{11}\) and \(x= -\sqrt{11}\)

Step 1: Identify the Equation

The given equation is \(x^{2}=11\). This a quadratic equation of the type \(x^2 = n\), where n is a positive number.

Step 2: Obtain the Square Root of Both Sides

Taking square roots for both sides of the equation will help to isolate the variable x. Remember to consider both the positive and negative roots. Hence, the solutions are \(x=\sqrt{11}\) and \(x=-\sqrt{11}\).

Step 3: Simplify the Solution

Since 11 is a prime number, it does not have any perfect square divisors (other than 1) and thus cannot be simplified any further. Therefore, the solutions remain at \(x=\sqrt{11}\) and \(x=-\sqrt{11}\).

Key Concepts

Understanding Square Roots

When we come across a quadratic equation like the one in our exercise, the square root plays a central role in finding the solution. A square root of a number 'n' is a value that, when multiplied by itself, gives 'n'. For instance, the square root of 9 is 3 because when you multiply 3 by itself (3 \times 3), the result is 9. In our exercise, solving for the square root of 11 involves identifying a number that, when squared, would result in 11. Since 11 is not a perfect square (like 4, 9, or 16), we represent the answer in radical form, which looks like \(\sqrt{11}\).

In general, all positive numbers have two square roots: a positive one and a negative one, because both \(+\sqrt{n}\) when squared and \(-\sqrt{n}\) when squared, will result in the original number 'n'. That’s why the equation \(x^2 = 11\) has two solutions: \(x = +\sqrt{11}\) and \(x = -\sqrt{11}\).

Dealing with Radical Expressions

Radical expressions contain roots, such as square roots or cube roots, and often students need to simplify these expressions to find a solution. Simplifying involves finding the prime factors of a number and then determining if any of those factors are perfect squares. Perfect squares can be taken out from under the radical sign, making the expression simpler.

However, in our example \(\sqrt{11}\), the number 11 is a prime number and does not have any perfect square factors. Therefore, the expression cannot be simplified further and the answer remains as a radical. Expressing solutions in radical form is essential when exact values are needed, and the number inside the radical isn't a perfect square. For students, it's important to get comfortable with radical expressions like \(\sqrt{11}\) because they frequently appear in higher-level mathematics and various applications.

Solving Quadratic Equations

A quadratic equation is any equation that can be written in the form \(ax^2 + bx + c = 0\), where 'a', 'b', and 'c' are constants, and 'a' is not zero. In the case of \(x^2 = 11\), it is a simpler form of a quadratic equation with 'a' being 1, 'b' being 0, and 'c' as -11. These types of equations are fundamental in algebra and appear across various fields of science and engineering.

To solve them, one common method is factoring, but when equations can’t be factored easily, we can use other methods such as completing the square, using the quadratic formula, or like in our example, taking the square root. When using the square root method for an equation in the form of \(x^2 = n\), we take the square root of both sides, and we always consider both the positive and negative roots. It's a straightforward technique that can provide solutions quickly for certain types of quadratic equations.