91Ó°ÊÓ

One of the fundamental concepts in high school algebra is the quadratic equation. This is an equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The equation given in this example, \(x^2 - 35 = 0\), is a simplified form of a quadratic equation where \(b = 0\) and \(c = -35\). Quadratic equations like these describe a parabola and have degrees of 2—meaning the highest power of the unknown \(x\) is 2. The solutions to these equations are often found by factoring, completing the square, using the quadratic formula, or through properties like the square root property. Each method caters to different forms and complexities of quadratic equations.

When attempting to solve a quadratic equation using the square root property, the first step is to isolate the quadratic term, which, in this case, is \(x^2\). This means we want \(x^2\) to be by itself on one side of the equation without any constants or coefficients.

For \(x^2 - 35 = 0\), isolating \(x^2\) requires moving the constant \(-35\) to the other side. We achieve this by adding 35 to both sides of the equation. This operation gives us \(x^2 = 35\).

Let's summarize the steps to isolate a quadratic term:

• Identify the quadratic term you need to isolate, usually \(x^2\).
• Eliminate constants or terms added or subtracted by performing the opposite arithmetic operation on both sides.
• Ensure the quadratic term is alone on one side of the equation.

After isolating \(x^2\), you're prepared to apply further techniques, like taking the square root, to solve the equation.

After isolating the quadratic term and applying the square root property, it's essential to simplify the result. The square root property gives us two potential answers for \(x\), \(x = \pm \sqrt{35}\), because taking the square root of a value can give both a positive and a negative outcome.

However, simplifying involves more than just stating these potential solutions. Ideally, we want to make the expression as simple as possible. But since \(\sqrt{35}\) is already in its simplest form—35 is not a perfect square and does not have any perfect square factors beyond 1—there's no further simplification possible.

This step ensures that the solutions are presented in a clear and concise format.

• Check if the radical can be simplified if the number under the square root is a perfect square.
• Write solutions clearly, usually in the format of \(x = \sqrt{a}\) and \(x = -\sqrt{a}\).

The ability to simplify results solidifies the process of solving quadratic equations, making sure the answer is both correct and cleanly presented.