Problem 70 - Elementary Algebra

Question

\(5 x^{2}=-35\)

Step-by-Step Solution

Answer

The solutions are \( x = i\text{√7} \) and \( x = -i\text{√7} \).

Step 1: Isolate the variable term

First, divide both sides of the equation by 5 to isolate the variable term: \[ 5x^2 = -35 \] Divide both sides by 5: \[ x^2 = -7 \]

Step 2: Solve for x

Take the square root of both sides to solve for x. Remember to consider both the positive and negative roots: \[ x = \frac{\text{±}\text{√-7}}{\underline{\phantom{xx}}} =\text{ullull ±}\text{i √7} \] Here, \( i \) represents the imaginary unit, since we have a negative number under the square root.

Key Concepts

Isolating Variable Terms

To solve any equation, one of the first steps is to isolate the variable term. This means we want to get our variable (in this case, x) by itself on one side of the equation.
In the given problem, we start with the equation:

• 5x^2 = -35

Because the variable term (x^2) is being multiplied by 5, we need to divide both sides of the equation by 5 to isolate x^2. This gives us:

• \( x^2 = -7 \)

Isolating variable terms is a useful strategy because it simplifies the equation, making it easier to solve for x. This step is a foundational skill in algebra and is used in a wide range of problems.
Remember that whatever operation you do to one side of the equation, you must also do to the other side.

Imaginary Numbers

In mathematics, we often encounter situations where we need to take the square root of a negative number. This is where imaginary numbers come in. By definition, the square root of -1 is represented as i, which is the imaginary unit.
So, when we have the equation:

• \( x^2 = -7 \)

and we need to solve for x, we take the square root of both sides. This gives us:

• \( x = \text{±}\text{i} √7 \)

Here, i is used to represent the square root of -1, allowing us to express the square root of a negative number.
Imaginary numbers are a fundamental part of complex numbers, which have both real and imaginary parts. They may seem abstract but are very useful in many fields of science and engineering.
When working with imaginary numbers, always remember that i is used to denote the square root of -1.

Square Roots

Taking the square root is an essential part of solving quadratic equations. When you have an equation like:

• \( x^2 = -7 \)

we need to take the square root of both sides to solve for x.
Normally, the square root of a positive number yields two results: a positive root and a negative root. However, with negative numbers, we get imaginary roots. So, you write:

• \( x = \text{±}\text{i} √7 \)

This means x can be either positive or negative i√7.
Understanding square roots is critical because they are used in a variety of mathematical contexts, from solving simple equations to more complex functions. Square roots help us find values that satisfy quadratic equations and are pivotal in the study of parabolas and other curves.
When taking square roots, always consider both the positive and negative solutions.