Question

\(x^2=100-y^2\)

Short answer

The solution to the equation \(x^2 = 100 - y^2\) is given by \(x = \sqrt{100 - y^2}\) and \(x = -\sqrt{100 - y^2}\).

Step-by-step solution

Identify the Form of the Equation

Recognize that the given equation \(x^2=100-y^2\) is in the form of \(a^2 - b^2\). This pattern can be factored into the form \((a + b)(a - b)\). In this case, \(a\) corresponds to \(x\) while \(b\) is the square root of 100 or 10.

Simplify the Equation

The next step is to simplify the equation by moving \(y^2\) to the left side of the equation, so it now reads \(x^2 + y^2 = 100\). This is done using the arithmetic operation of addition.

Solving for x

The final step involves isolating \(x\) on one side of the equation. Subtract \(y^2\) from both sides of the equation to get \(x^2 = 100 - y^2\). Then take the square root of both sides, to get \(x = \sqrt{100 - y^2}\). Remember, that there would be two solutions, \(x = \sqrt{100 - y^2}\) and \(x = -\sqrt{100 - y^2}\), due to the property of square roots.

Key concepts

Quadratic Equations

A quadratic equation is an equation that can be expressed in the standard form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). These equations are fundamental in algebra and appear in many mathematical problems. The key characteristic of a quadratic is the squared term, \(x^2\).

Unlike linear equations, which graph as straight lines, quadratic equations graph as parabolas. Understanding their structure helps in solving them in different ways, such as factoring, completing the square, or using the quadratic formula. In the equation presented, \(x^2 = 100 - y^2\), the term \(x^2\) underlines its quadratic nature.

A key skill is recognizing how to manipulate quadratic equations to find solutions, like rearranging or factoring, allowing for easier problem-solving and solution finding.

Factoring

Factoring is a valuable method used to break down expressions into products of simpler factors. This technique is particularly useful for solving quadratic equations. Factorable quadratics typically show a pattern or structure which helps to simplify the problem-solving process.

One popular pattern is the difference of squares, which follows the identity \(a^2 - b^2 = (a + b)(a - b)\). This method is useful to simplify expressions quickly. In the original exercise, the equation \(x^2 = 100 - y^2\) can be visualized as a difference of two squares by identifying the squared terms \(x^2\) and \(y^2\) against a constant squared term 100 (as \(10^2\)).

By recognizing and applying the difference of squares, you can efficiently rearrange the equation into a factorable format. This reduces complexity and assists in finding the variable values easily.

Pythagorean Theorem

The Pythagorean Theorem is a cornerstone in geometry, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides: \(a^2 + b^2 = c^2\). This theorem connects algebra and geometry, providing insight into the relationship between the lengths of sides in right triangles.

In the exercise, when the equation \(x^2 = 100 - y^2\) is rearranged to \(x^2 + y^2 = 100\), it takes on a form resembling the Pythagorean Theorem. Here, \(x\) and \(y\) can be viewed as the legs of a right triangle whose hypotenuse has a length of 10, as shown by the constant 100 being \(10^2\).

This illustrates the beauty of mathematics where algebraic manipulation meets geometric interpretation, offering a comprehensive approach to solving problems in diverse scenarios.