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Quadratic equations are polynomial equations of the second degree, typically in the form of \[ax^2 + bx + c = 0\]. These equations can have various types of solutions, including real and complex numbers. A key feature of quadratic equations is the squared term (\(x^2\)). Understanding how to solve these equations is crucial because they appear in many areas of mathematics, science, and engineering. Some common methods to solve them include factoring, completing the square, and using the quadratic formula (\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)).

Solving equations means finding the values of the unknown variable that satisfy the equation. For quadratic equations, the steps to solve them often start with moving terms around to isolate the term with the variable on one side.

• Step 1: Rearrange the equation to move the constant term to the other side.
• Step 2: Simplify to isolate the squared term (often involving division).
• Step 3: Take the square root of both sides to solve for the variable.

In our example, the process is as follows: Add 25 to both sides to get \(36a^2 = 25\). Then, divide both sides by 36, resulting in \(a^2 = \frac{25}{36}\). Finally, take the square root of both sides to find \(a = \pm \frac{5}{6}\). Remember to consider both the positive and negative square roots.

The square root is a value that, when multiplied by itself, gives the original number. For instance, \( \sqrt{36} = 6\), because \(6 \times 6 = 36\). When solving quadratic equations, taking the square root of both sides is often the final step to isolate the variable. Importantly, every positive number has two square roots: a positive one and a negative one. Therefore, \(x^2 = k\) implies \(x = \pm \sqrt{k}\).
In our problem, after isolating \(a^2 = \frac{25}{36}\), taking the square root of both sides gives \(a = \pm \frac{5}{6}\). This step converts a squared term into its linear form, providing the solutions to the equation.
Understanding how to correctly handle square roots is crucial for solving quadratic equations and ensuring that all possible solutions are found.