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To find the x-intercepts of a quadratic equation, like the one given in the problem \( y = x^2 - 3x - 10 \), we set \( y \) equal to zero and solve for \( x \). The x-intercepts are the points where the graph of the equation crosses or touches the x-axis. This is important because it tells you where the quadratic function equals zero.

There are several methods to solve for the x-intercepts, including:

• Factoring: Expressing the quadratic equation in a product form, e.g., \((x-5)(x+2) = 0\), helps find the solutions by setting each factor equal to zero. This solves to \( x = 5 \) and \( x = -2 \).
• Quadratic Formula: This formula is used when factoring is complex. It's written as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) for any quadratic equation \( ax^2 + bx + c = 0 \).
• Completing the Square: This involves rearranging the equation to make it a perfect square trinomial, then solving for \( x \).

In the given quadratic, factoring was used as it efficiently provided the x-intercepts at \( x = 5 \) and \( x = -2 \). Each solution represents a point where the parabola intersects the x-axis.

Finding the y-intercept of a quadratic equation involves setting \( x \) to zero, which simplifies the process. In the example \( y = x^2 - 3x - 10 \), substituting \( x = 0 \) gives \( y = -10 \). This value represents the y-intercept, located on the y-axis where the graph of the quadratic touches or crosses.

Understanding this concept provides a clear picture of where the parabola intersects the y-axis. Since quadratic equations can extend infinitely in either direction along the x-axis, the y-intercept offers a fixed point that simplifies the graphing process.

• The y-intercept is a constant that is easy to find. By directly substituting \( x = 0 \), we solve for \( y \), which results in \( y = -10 \).
• This point is crucial in understanding the general direction and position of the parabola relative to the y-axis.

The y-intercept essentially indicates the beginning position of the parabola on the y-axis, which contributes to a fuller understanding of the function's graph.

Factoring quadratics is an essential skill when solving quadratic equations. It involves rewriting the equation in a form that makes it easier to identify the solutions. In the expression \( y = x^2 - 3x - 10 \), we can factor it as \((x-5)(x+2)\) by finding two numbers that multiply to \(-10\) and add to \(-3\).

To factor effectively:

• Look for two numbers that multiply to the constant term (here \(-10\)) and add to the linear coefficient (here \(-3\)).
• In this problem, \(-5\) and \(2\) meet these conditions. Thus, rewriting as \((x - 5)(x + 2) = 0\) reveals the roots of the equation easily.

Factoring aids in breaking down complex equations into simpler terms, making it easier and quicker to find solutions. Understanding this method:

• It significantly reduces the effort in solving the equation compared to methods such as the quadratic formula.
• It requires practice to identify the right factors, especially for more complex quadratics.

Mastering factoring not only simplifies solving quadratics but also enhances overall algebraic skills, essential for tackling various mathematical problems.