91Ó°ÊÓ

A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 times 3 equals 9. In mathematical notation, we represent the square root using the symbol \(\sqrt{}\). For instance, \(\sqrt{9} = 3\).

In the context of solving quadratic equations, taking the square root is a common step. For the given problem, we start by isolating the square root on both sides of the equation: \[\sqrt{\left(y+\frac{3}{4}\right)^{2}} = \sqrt{\frac{17}{16}}\].

This step is crucial because it simplifies the equation. The square and the square root cancel each other out on the left side, leaving us with \(y + \frac{3}{4}\). On the right side, we calculate the square root of \(\frac{17}{16}\), which simplifies to \(\frac{\sqrt{17}}{4}\). Thus, the equation becomes: \[ y + \frac{3}{4} = \pm \frac{\sqrt{17}}{4}\].

When solving equations involving square roots, we must account for both positive and negative roots. This principle arises because both positive and negative values, when squared, yield a positive result. For example, both \(2^2 = 4\) and \((-2)^2 = 4\).

In our problem, we encounter \(\pm \frac{\sqrt{17}}{4}\). This notation tells us that there are two possible solutions — one where \(y + \frac{3}{4} = \frac{\sqrt{17}}{4}\) and another where \(y + \frac{3}{4} = -\frac{\sqrt{17}}{4}\).

The plus-minus symbol is crucial because it captures both potential roots in a single expression.

By splitting the equation into two separate cases, we address each root individually: one for the positive value and another for the negative value. This approach ensures we find all possible solutions.

Quadratic equations, which take the form \(ax^2 + bx + c = 0\), often have two solutions. These solutions can be found through various methods, such as factoring, completing the square, or using the quadratic formula.

In our specific case, we solve the quadratic equation by taking the square root and then isolating the variable \(y\).

We first simplify the equation to \(y + \frac{3}{4} = \pm \frac{\sqrt{17}}{4}\). Next, we split this into two scenarios: one for the positive root and another for the negative root.

For the positive root, we subtract \(\frac{3}{4}\) from both sides to get: \[ y = \frac{\sqrt{17}}{4} - \frac{3}{4} = \frac{\sqrt{17} - 3}{4}\].

Similarly, for the negative root, we subtract \(\frac{3}{4}\) from both sides to get: \[ y = -\frac{\sqrt{17}}{4} - \frac{3}{4} = \frac{-\sqrt{17} - 3}{4}\].

These steps ensure that we have explored both scenarios, providing a complete solution set for the quadratic equation.