91Ó°ÊÓ

When we solve equations, particularly quadratic equations, our primary goal is to find the values of the unknowns that make the equation true. In these equations, variables like \(x\) are involved, and operations such as multiplication, division, and exponentiation are used. The quadratic equation in the original exercise is \(5x^2 = 30\). This equation requires us to find the value of \(x\) that makes both sides of the equation equal.

The process begins with simplifying the equation step-by-step until the variable \(x\) is isolated. For quadratic equations, this often involves dividing both sides by any coefficient that is attached to \(x^2\). By doing so, we ensure that we are left with a simpler equation, making the solution more manageable.

The square root method is a technique for solving quadratic equations that are in the form \(x^2 = \text{constant}\). Once an equation is simplified to this form, the next step is to solve for \(x\) by taking the square root of both sides. In our original solution, we had \(x^2 = 6\).

By applying the square root method, we determine \(x\) by finding both the positive and negative square roots of the constant. Hence, \(x = \pm\sqrt{6}\).

• "\( \pm \)" signifies both the positive and negative roots because squaring either positive or negative numbers results in the same positive value.
• Knowing both roots is essential as quadratic equations can have two solutions.

This method is especially convenient for equations that don't factor easily, providing an efficient way to determine the solutions.

Simplifying expressions is a key step in solving equations, including quadratic ones. It involves reducing the equation to its simplest form to make calculations easier and solutions clearer. In our initial equation, \(5x^2 = 30\), simplification was performed by:

• Dividing both sides by 5, leading to \(x^2 = 6\).
• Eliminating the coefficient 5 from \(x^2\), isolating \(x^2\) on one side of the equation.

Each simplification step reduces the complexity of the equation and helps focus on the essential part that needs solving. This way, when arriving at a simplified expression, one can readily apply further techniques, like the square root method, to find the solution. Simplifying an expression not only streamlines calculations but also minimizes errors during the problem-solving process.