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Intercepts are fundamental points where a curve meets or crosses the axes in a coordinate system. For a given circle like ours, these intercepts tell us where the circle touches the x-axis or y-axis.

• x-intercepts: Occur where the graph meets the x-axis. This means the y-value is zero.
• y-intercepts: Happen where the graph touches the y-axis, which means the x-value is zero.

To find intercepts, set the other variable to zero and solve the remaining equation. In our example, setting \(y = 0\) gives the x-intercepts, while \(x = 0\) yields the y-intercepts. The intercepts include both points such as

• (0, 0) - where the circle crosses both axes at the origin
• (-5, 0) - an additional x-intercept.

Similarly, the y-intercept at (0, 6) shows another crossing point on the y-axis.

Quadratic equations are polynomial equations of degree 2, generally in the form \( ax^2 + bx + c = 0 \). They are crucial in finding intercepts since intercept calculations often involve solving such equations.

In our circle's equation, setting \( y = 0 \) gives us the quadratic \( x^2 + 5x = 0 \), and setting \( x = 0 \) results in \( y^2 - 6y = 0 \). Each of these is a simpler form of the original equation that we can solve to get intercepts.

Every quadratic equation can be solved by:

• Factoring: Breaks the equation into less complex expressions.
• Completing the square: Useful for finding maximum or minimum values.
• Using the quadratic formula: This method is a guarantee when factoring seems difficult.

For the x-intercepts, the factoring approach works smoothly where we found \( x(x + 5) = 0 \). Similarly, for the y-intercepts, \( y(y - 6) = 0 \), both quadratic forms pointing to simple solutions by setting each factor equal to zero.

Factoring is a mathematical technique used to simplify equations and solve quadratic equations easier. By expressing a polynomial as a product of its factors, solving becomes straightforward.

When given an equation like \( x^2 + 5x = 0 \) from the x-intercepts search, factoring involved identifying terms that multiply to form the original equation.

The steps to factor:

• Identify common terms: In our case, \( x \) was common, giving us \( x(x + 5) = 0 \).
• Set each factor equal to zero: This gives possible solutions which intersect the axes. Here, it gives us \( x = 0 \) and \( x = -5 \).

Similarly, with \( y^2 - 6y = 0 \), the factorization yields \( y(y - 6) = 0 \), resulting in solutions \( y = 0 \) and \( y = 6 \). This process is a direct way to solve for intercepts and is often quicker than alternative methods like completing the square or using the quadratic formula.