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Quadratic equations are equations that involve terms with variables raised to the power of two, generally expressed in the standard form as \(ax^2 + bx + c = 0\). An important aspect of solving these equations is recognizing that they can have up to two solutions.

To effectively solve a quadratic equation, you first need to determine the method that best applies. This might include factoring, using the quadratic formula, or completing the square. When a quadratic equation is set to zero, as it often is, the objective is to find the values of the variable that make the equation true.

The zeros of the quadratic function correspond to the solutions of the equation, providing the values where the graph of the equation crosses the x-axis. For the given exercise, the equation \(r^2 - 49 = 0\) is straightforward since the coefficient of \(r\) is zero. Here, factoring is the most efficient approach.

Factoring is a technique used to solve quadratic equations by expressing the equation as a product of binomial expressions. The main goal is to rearrange the quadratic equation into the form \((x - p)(x - q) = 0\), where \(p\) and \(q\) are the solutions.

In the exercise \(r^2 - 49 = 0\), the equation can be rewritten using a factoring method. We identify that \(49\) is a perfect square of \(7\), allowing us to express the quadratic as \((r + 7)(r - 7) = 0\).

To solve after factoring, we set each factor equal to zero:

• \(r + 7 = 0\)
• \(r - 7 = 0\)

Each equation yields a solution for \(r\), illustrating the roots as \(r = -7\) and \(r = 7\). Factoring is powerful for solving quadratic equations when the terms can be easily decomposed into simpler expressions.

Perfect square factors are numbers that result from squaring an integer. Recognizing perfect squares can simplify solving quadratic equations because they allow the equation to be easily factored.

In the given exercise, \(49\) is identified as a perfect square because \(49 = 7^2\). This crucial insight lets us factor the quadratic equation \(r^2 - 49 = 0\) using the difference of squares method: \((r + 7)(r - 7) = 0\).

This method relies on the identity \(a^2 - b^2 = (a+b)(a-b)\), where \(a\) represents \(r\) and \(b\) represents \(7\). These perfect square factors not only simplify the factoring process but also directly lead to straightforward solutions. By recognizing perfect squares, solving quadratic equations can become an easier and more efficient task.