91Ó°ÊÓ

In mathematics, quadratic equations form a significant part of algebra and have various applications. A quadratic equation typically is of the form: \(ax^2 + bx + c = 0\). This equation has a degree of 2, which means the highest power of the variable \(x\) is 2.
Solving a quadratic equation means finding the value(s) of \(x\) that make the equation true. These values are called roots or solutions.

There are multiple methods to solve quadratic equations which include:

• Factoring: This involves expressing the quadratic equation as a product of its factors.
• Completing the Square: This method involves creating a perfect square trinomial on one side of the equation.
• The Quadratic Formula: Given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), this formula can solve any quadratic equation.

In the given exercise, we rearrange and solve using the square roots, since it is a simple equation with no linear component. If the equation is in the form \(x^2 = k\), you can directly take the square root of both sides to find \(x\). This leads us to the next core concept.

The x-axis intersection of a curve can be thought of as the points where the graph crosses the x-axis. For a parabola, which is a U-shaped graph, the intersection points are known as the roots or the solutions of the quadratic equation.

To find these intersections for the function \(y = 4 - x^2\), you set \(y\) equal to zero because you want to find where the graph touches or crosses the x-axis at zero height.
This means solving the equation \(4 - x^2 = 0\).

At the x-axis intersection points, the y-value is always zero resulting in the coordinate form of intersection points as \((x_1, 0)\) and \((x_2, 0)\).
In our specific example, solving the equation results in two x-axis intersections at \(x = 2\) and \(x = -2\), yielding the intersection points as \((2, 0)\) and \((-2, 0)\).
These points tell us where the parabola \(y = 4 - x^2\) is zero.

Square roots are an essential part of solving equations, especially quadratics. The square root of a number \(n\) is another number \(m\) that, when multiplied by itself, gives \(n\). This is expressed as \(m^2 = n\) or \(m = \sqrt{n}\).

One important thing to remember about square roots is that every positive number has two square roots: one positive and one negative. For example, the square roots of 4 are both 2 and -2, since \(2^2 = 4\) and \((-2)^2 = 4\).

In solving quadratic equations like \(x^2 = 4\), we use the property of square roots to find values of \(x\). Taking the square root of both sides, we get \(x = \sqrt{4}\), resulting in \(x = 2\) or \(x = -2\).
This ability to take the square root is critical in finding roots of quadratic equations where the term in question can be expressed as a perfect square.